Medical Imaging Signals and Systems
Jerry L. Prince
Johns Hopkins University
August 20, 2009
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 1 / 412
AcknowledgementsThese notes are intended to be used with the thetextbook:
▶ Jerry L. Prince and Jonathan M. Links,“Medical Imaging Signals and Systems,” UpperSaddle River: Pearson Prentice Hall, 2006.
Images and figures lacking a specific bibliographiccitation are either taken from this book and thecopyright is owned by Pearson Prentice Hall or werehand drawn by Jerry Prince.
Images and figures with specific bibliographiccitations are used with permission of the copyrightholder.
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 2 / 412
Outline
Outline I
1 Introduction to Medical Imaging Systems
2 Multidimensional Signal Processing
3 Image Quality
4 Physics of Radiography
5 Projection Radiography
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 3 / 412
Outline
Outline II6 Computed Tomography
7 Physics of Nuclear Medicine
8 Planar Scintigraphy
9 Emission Tomography
10 Ultrasound Physics
11 Ultrasound Imaging
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Outline
Outline III12 Physics of Magnetic Resonance
13 Magnetic Resonance Imaging
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Introduction to Medical Imaging Systems
1 Introduction to Medical Imaging SystemsOverall PerspectivePossible objectivesSignals Examples
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Introduction to Medical Imaging Systems Overall Perspective
Overall PerspectiveCourse breakdown
▶ 1/3 physics▶ 1/3 instrumentation▶ 1/3 signal processing
Understand systems from a “signals” viewpoint:
input signal→ system or process→ output signal
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Introduction to Medical Imaging Systems Overall Perspective
A Signal Example
ExampleInput signal: �(x , y) is the linear attenuationcoefficient for x-rays
Process (integration over x variable):
g(y) =
∫�(x , y)dx
Output signal: g(y)
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Introduction to Medical Imaging Systems Possible objectives
Possible objectivesunderstand “noise” or “artifacts” created by system
understand “contrast” in input and output
process output to create a “picture” of input
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Introduction to Medical Imaging Systems Signals Examples
Examples of Signals in Medical Imaging�(x , y , z), linear attenuation coefficient in x-rays
h(x , y , z), CT numbers in computed tomography
A(x , y , z), radioactivity in nuclear medicine
Chest X-ray Abdominal CT Cardiac SPECT
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Introduction to Medical Imaging Systems Signals Examples
More ExamplesPD(x , y , z), proton density in MRI imaging
T1(x , y , z), longitudinal relaxation time in MRI
T2(x , y , z), transverse relaxation time in MRI
PD-weighted T2-weighted T1-weighted
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Introduction to Medical Imaging Systems Signals Examples
More ExamplesR(x , y , z), reflectivity in ultrasound imaging
vR(x , y , z), range component of velocity in Dopplerultrasound
11-week Embryo Fetus Heart
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Multidimensional Signal Processing
2 Multidimensional Signal ProcessingMultidimensional SignalsDelta FunctionsSystemsFourier TransformRect and SincHankel TransformSamplingAliasingArea Detectors
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Multidimensional Signal Processing Multidimensional Signals
1D, 2D, and 3D SignalsA 1D signal is:
▶ f (t), a function of one variable, or▶ a waveform, or▶ a graph (a collection of points in a 2D space)
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Multidimensional Signal Processing Multidimensional Signals
A 2D signal is:▶ f (x , y), a function of two variables, or▶ an image, or▶ a graph (a collection of points in a 3D space)
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Multidimensional Signal Processing Multidimensional Signals
A 3D signal is:▶ f (x , y , z), a function of three variables, or▶ a “volumetric image,” or▶ a graph (a collection of points in a 4D space)
We focus (mostly) on 2D signals in this course
Separable signals:▶ f (x , y) = f1(x)f2(y)▶ f (x , y , z) = f1(x)f2(y)f3(z)
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Multidimensional Signal Processing Delta Functions
Delta FunctionsThe 1D delta or impulse “function” is defined bytwo properties:
�(x) = 0 , x ∕= 0∫∞−∞ f (x)�(x)dx = f (0)
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Multidimensional Signal Processing Delta Functions
Properties of the Delta FunctionThe area of �(x) is unity∫ ∞
−∞�(x)dx = 1
A 2D delta function �(x , y) is defined by
�(x , y) = 0 , (x , y) ∕= 0∫∞−∞∫∞−∞ f (x , y)�(x , y)dx dy = f (0, 0)
A 3D delta function is analogous.
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Multidimensional Signal Processing Delta Functions
More PropertiesProperties of delta functions:
�(−x) = �(x) even
�(x , y) = �(x)�(y) separable∫ ∞−∞
f (�)�(� − x)d� = f (x) sifting
2D sifting property∫ ∞−∞
∫ ∞−∞
f (�, �)�(� − x , � − y)d�d� = f (x , y)
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Multidimensional Signal Processing Delta Functions
Point Source Modeldelta function models a point source
▶ metal bead in x-ray▶ vial of radioactivity in nuclear medicine▶ vitamin E pill in magnetic resonance imaging▶ small bubble or microcalcification in ultrasound
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Multidimensional Signal Processing Systems
Transformations of SignalsComponents of a transformation:
▶ Input: f▶ System: ℋ[⋅]▶ Output: g
The impulse response or point spread function dueto an impulse at (�, �) is
h(x , y ; �, �) = ℋ[�(x − �, y − �)]
h(x , y ; �, �) is a 2D signal parameterized by a 2Dvector
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Multidimensional Signal Processing Systems
A linear system satisfies:
ℋ[w1f1 + w2f2] = w1ℋ[f1] + w2ℋ[f2]
for all signals f1 and f2 and weights w1 and w2.
A linear system satisfies the superposition integral
g(x , y) =
∫ ∞−∞
∫ ∞−∞
h(x , y ; �, �)f (�, �)d�d�
We model most medical imaging systems as linear.
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Multidimensional Signal Processing Systems
Shift-Invariant SystemsA system is shift-invariant is
g(x − x0, y − y0) = ℋ[f (x − x0, y − y0)]
for every (x0, y0) and f (⋅, ⋅).
A linear shift-invariant (LSI) system yields
h(x , y ; �, �)→ h(x − �, y − �)
[Watch out for abuse of notation]
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Multidimensional Signal Processing Systems
Convolution IntegralAn LSI system satisfies the convolution integral
g(x , y) =
∫ ∞−∞
∫ ∞−∞
h(x − �, y − �)f (�, �)d�d�
which is abbreviated as
g(x , y) = h(x , y) ∗ f (x , y)
We model most medical imaging systems as LSI
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Multidimensional Signal Processing Fourier Transform
LSI Systems and Complex ExponentialsA 2D complex exponential signal is
e j2�(ux+vy) = e j2�uxe j2�vy i.e., separable
wheree j2�ux = cos 2�ux + j sin 2�ux
The response of an LSI system to
f (x , y) = e j2�(ux+vy)
isg(x , y) = H(u, v)e j2�(ux+vy)
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Multidimensional Signal Processing Fourier Transform
The function
H(u, v) =
∫ ∞−∞
∫ ∞−∞
h(x , y)e−j2�(ux+vy)dxdy
H(u, v) is called the Fourier transform of h(x , y).
The inverse Fourier transform of H(u, v) is
h(x , y) =
∫ ∞−∞
∫ ∞−∞
H(u, v)e+j2�(ux+vy)dudv
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Multidimensional Signal Processing Fourier Transform
Magnitude of the 2D Fourier Transform
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Multidimensional Signal Processing Fourier Transform
Comments on the Fourier TransformNotation:
F (u, v) = ℱ{f }
=
∫ ∞−∞
∫ ∞−∞
f (x , y)e−j2�(ux+vy)dxdy
f (x , y) = ℱ−1{F}
=
∫ ∞−∞
∫ ∞−∞
F (u, v)e+j2�(ux+vy)dudv
e j2�(ux+vy) is a complex sinusoid “oriented” in the(u, v) direction
2�ux has units of radians
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Multidimensional Signal Processing Fourier Transform
⇒ ux is unitless
⇒ x has units of length, e.g., cm or mm
⇒ u has units of inverse length, e.g., cm−1 ormm−1.
u is referred to as (cyclic) spatial frequency
The 1D Fourier transform pair is given by
F (u) =
∫ ∞−∞
f (x)e−j2�uxdx
f (x) =
∫ ∞−∞
F (u)e+j2�uxdu
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Multidimensional Signal Processing Fourier Transform
Properties of the Fourier Transform[Refer to text for complete list]
Linearity:
ℱ{w1f1 + w2f2} = w1F1 + w2F2
Scaling:
ℱ{f (�x , �y)} =1
∣��∣F (
u
�,v
�)
Shifting:
ℱ{f (x − �, y − �)} = F (u, v)e−j2�(u�+v�)
ℱ{f (x , y)e+j2�(�x+�y)} = F (u − �, v − �)
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Multidimensional Signal Processing Fourier Transform
Convolution:
ℱ{f1 ∗ f2} = F1F2
Correlation:
ℱ{∫ ∞−∞
∫ ∞−∞
f1(�, �)f ∗2 (x + �, y + �)d�d�
}= F1(u, v)F ∗2 (u, v)
Separable input: If f (x , y) = f1(x)f2(y) then
ℱ{f (x , y)} = F1(u)F2(v)
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Multidimensional Signal Processing Fourier Transform
Parseval’s theorem:∫ ∞−∞
∫ ∞−∞∣f (x , y)∣2dxdy
=
∫ ∞−∞
∫ ∞−∞∣F (u, v)∣2dudv
Product:
ℱ{f1(x , y)f2(x , y)} = F1(u, v) ∗ F2(u, v)
Impulse:ℱ{�(x , y)} = 1
Constant:ℱ{1} = �(u, v)
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Multidimensional Signal Processing Fourier Transform
Sinusoid (1D):
ℱ{sin 2�u0x} =1
2j[�(u − u0)− �(u + u0)]
ℱ{cos 2�u0x} =1
2[�(u − u0) + �(u + u0)]
Sinusoid (2D):
ℱ{sin 2�(u0x + v0y)}
=1
2j[�(u − u0, v − v0)− �(u + u0, v + v0)]
ℱ{cos 2�(u0x + v0y)}
=1
2[�(u − u0, v − v0) + �(u + u0, v + v0)]
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Multidimensional Signal Processing Rect and Sinc
Rect and SincRect function: (“gate” or “pedestal”)
rect(x) =
{1 ∣x ∣ ≤ 1/20 otherwise
Sinc function:
sinc(x) =sin �x
�x
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Multidimensional Signal Processing Rect and Sinc
Fourier transform relationship:
ℱ{rect(x)} = sinc(u)
x
sinc( )x
1
−1 1 2 3 40−2−3−4
rect( )x
x1 / 2−1 / 2
1
0
(a) (b)
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Multidimensional Signal Processing Hankel Transform
RotationRotation:
f�(x , y) = f (x cos � − y sin �, x sin � + y cos �)
Fourier transform rotates also
ℱ2D(f�)(u, v)
= F (u cos � − v sin �, u sin � + v cos �)
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Multidimensional Signal Processing Hankel Transform
Circular Symmetry2D signal is circularly symmetric if
f�(x , y) = f (x , y) , for every �
ℱ2D(f�)(u, v) is also circularly symmetric
f (x , y) and F (u, v) are functions of radii only
f (x , y) = f (r)
andF (u, v) = F (q)
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Multidimensional Signal Processing Hankel Transform
Hankel TransformFourier transform of circularly symmetric objects isdescribed by the Hankel transform
F (q) = 2�
∫ ∞0
f (r)J0(2�qr) r dr
J0(r) is zero-order Bessel function of the first kind
J0(r) =1
�
∫ �
0
cos(r sin�) d�
Example pair:
ℋ{exp{−�r 2}} = exp{−�q2}Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 38 / 412
Multidimensional Signal Processing Sampling
Sampling
x
y
∆x
∆y
∆x
∆y
x
y
Point sampling:
f [m, n] = f (mΔx , nΔy)
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Multidimensional Signal Processing Sampling
Impulse TrainsImpulse train or comb or shah function:
comb(x) =∞∑
n=−∞�(x − n)
Fourier transform relationship
ℱ{comb(x)} = comb(u)
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Multidimensional Signal Processing Sampling
Sampling FunctionThe sampling function:
�s(x ; Δx) =∞∑
n=−∞�(x − nΔx)
Impulse scaling property:
�(ax) =1
∣a∣�(x)
Relation to shah/comb function:
�s(x ; Δx) =1
Δxcomb
( x
Δx
)Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 41 / 412
Multidimensional Signal Processing Sampling
Sampling Model (see text for 2D)Sampled signal
fs(x) = f (x)�s(x ; Δx)
fs(x) contains the same information as
f [k] = f (kΔx)
Fourier transform of fs(x):
Fs(u) = F (u) ∗ ℱ{�s(x ; Δx)}
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Multidimensional Signal Processing Sampling
Sampled SpectrumFourier transform of sampling function:
ℱ{�s(x ; Δx)} = comb(Δxu)
=1
Δx
∞∑−∞
�(u − n
Δx)
Sampled spectrum is therefore:
Fs(u) =1
Δx
∞∑−∞
F (u) ∗ �(u − n
Δx)
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Multidimensional Signal Processing Sampling
Sampled Spectrum in 2D
u
vF u v( , )
V
−V
U−U
v
u
F u vs ( , )
(a) (b)
1 2/ ∆y
−1 2/ ∆y
−
1
2∆x
1 / ∆y
−1 / ∆y
−1 / ∆x 1 / ∆x
U−U
V
−V
1
2∆x
v
u
F u vs ( , )
−1 /∆y
−1 / ∆x 1 / ∆x
1 /∆y
U
−V
V
−U
aliasing
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Multidimensional Signal Processing Sampling
Sampling TheoremThe (spatial) sampling frequency is:
us =1
Δx
Let U be the highest frequency in F (u).
Then sampled spectra do not overlap if
us > 2U
2U is called the Nyquist rate
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Multidimensional Signal Processing Aliasing
AliasingAliasing occurs if us < 2U .
▶ Overlapping sampled spectra.▶ Corruption of high frequencies▶ Artifacts are high frequency patternsv
u
F u vs ( , )
−1 /∆y
−1 / ∆x 1 / ∆x
1 /∆y
U
−V
V
−U
aliasing
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Multidimensional Signal Processing Aliasing
Anti-aliasing FiltersSuppose:
▶ us = 1/Δx▶ Highest frequency in f (x) is U .
Filter f (x):▶ before sampling▶ Use low pass filter with cutoff frequency us/2.
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Multidimensional Signal Processing Area Detectors
Area Detector AnalysisShape of detector: p(x) [maybe rect(x/D)]
Area detector sampling model:
fs(x) = [p(x) ∗ f (x)]�s(x ; Δx)
Fourier domain:
Fs(u) = [P(u)F (u)] ∗ comb(Δxu)
= [P(u)F (u)] ∗ 1
Δx
∞∑n=−∞
�(u − n
Δx
)
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Image Quality
3 Image QualityBasic NotionsContrastResolutionNoiseArtifactsAccuracy
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Image Quality Basic Notions
What is Quality?What makes a good medical image?
▶ physics-oriented answer:faithful representation of the truth
▶ task-oriented answer:discrimination of healthy vs. diseasedtissues
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Image Quality Basic Notions
Measures of QualityPhysics-oriented issues:
▶ contrast, resolution▶ noise, artifacts, distortion▶ accuracy
Task-oriented issues:▶ sensitivity, specificity▶ diagnostic accuracy
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Image Quality Contrast
Contrast or ModulationSinusoidal image brightness function:
f (x , y) =fmax + fmin
2
+fmax − fmin
2sin(2�u0x)
Contrast = modulation =
mf =amplitude
average=
fmax − fmin
fmax + fmin
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Image Quality Contrast
Sinusoidal Signals with Different Contrast
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Image Quality Contrast
Sinusoid Input/Output in a Linear SystemInput: (assume f ≥ 0)
f (x , y) = A + B sin(2�u0x)
Assume impulse response h(x , y) is real
Output:
g(x , y) = H(0, 0)A
+ ∣H(u0, 0)∣B sin[2�u0x + ∠H(u0, 0)]
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Image Quality Contrast
Contrast Change in a Linear SystemInput contrast: mf = B/A
Output contrast:
mg =∣H(u0, 0)∣BH(0, 0)A
=∣H(u0, 0)∣H(0, 0)
mf
A B+
A B-
A
A B H u+ | ( , )|0
A B H u- | ( , )|0
mB
Af = m
B
AH ug = | ( , )|0
medical
imaging
system
input f x y( , ) output g x y( , )
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Image Quality Contrast
Modulation Transfer FunctionModulation transfer function:
MTF(u) =mg
mf=∣H(u, 0)∣H(0, 0)
spatial frequency u
0 8. mm-1
0
1 0.
MTF( )u
0 6.0 2. 0.4 1 0. 1 2. 1.4
0 5.
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Image Quality Contrast
More on Modulation Transfer FunctionGeneral case
MTF(u, v) =∣H(u, v)∣H(0, 0)
MTF is partial characterization of real system
Rule of thumb:▶ ∣H(u, v)∣ holds 1/8 of info▶ ∠H(u, v) hold 7/8 of info
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Image Quality Contrast
Contrast is Related to Resolution
decreasing contrast
MTF
outp
ut si
gna
l
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Image Quality Contrast
Local ContrastNon-sinusoidal signals: identify
▶ target intensity: ft▶ background intensity: fb
Local contrast:
C =ft − fbfb
Optional:▶ C (%) = C × 100%▶ C (abs) = ∣C ∣
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Image Quality Resolution
Resolution: Bar Phantom
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Image Quality Resolution
Bar Phantom Properties50% duty cycle
Material depends on modality▶ metal or plexiglass bars▶ tubes of radioactivity
resolution defined as the highest line density suchthat lines can be distinguished
units: line pairs (lp) per distance▶ gamma camera: 2–3 lp/cm▶ CT: 2 lp/mm▶ chest x-ray: 6–8 lp/mm
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Image Quality Resolution
Resolution: Line ResponseLine “function”:
f (x , y) = �(x)
Line response:
l(x) = S{�(x)} =
∫ ∞−∞
h(x , �)d�
Relation to MTF:
MTF(u) =∣L(u)∣L(0)
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Image Quality Resolution
Resolution: Line Separation
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Image Quality Resolution
Resolution: FWHMFull Width at Half Maximum (FWHM)
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Image Quality Noise
Noise: Random VariablesTypical imaging model:
g(x , y) = f (x , y) ∗ h(x , y) + N(x , y)
N(x , y) is noise
N(x , y) is a random variable at each (x , y)
N(x , y) could be continuous or discrete
Probability Distribution Function (PDF)
PN(�) = Pr[N ≤ �]
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Image Quality Noise
Continuous Random VariablesProbability density function (pdf):
pN(�) =dPN(�)
d�
Mean:
�N =
∫ ∞−∞
�pN(�)d�
Variance:
�2N =
∫ ∞−∞
(� − �)2pN(�)d�
Standard deviation:
�N =√�2N
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Image Quality Noise
Gaussian Random Variablepdf
pN(�) =1√
2��2e−(�−�)2/2�2
mean:�N = �
variance:�2N = �2
standard deviation:
�N = �
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Image Quality Noise
Discrete Random VariablesProbability mass function (PMF):
pN(�i) = Pr[N = �i ]
Mean:�N =
∑all �i
�ipN(�i)
Variance:
�2N =
∑all �i
(�i − �N)2pN(�i)
Standard deviation:
�N =√�2N
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Image Quality Noise
Poisson Random VariablePMF
pN(k) =ake−a
k!, for k = 0, 1, . . .
mean:�N = a
variance:�2N = a
standard deviation:
�N =√a
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Image Quality Noise
Sum of Independent Random VariablesLet N and M be joint random variables
Let Q = N + M
Then�Q = �N + �M
If N and M are independent then
�2Q = �2
N + �2M
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Image Quality Noise
Images Degrade with NoiseNoise level depends on modality and acquisitionparameters
Fast imaging is almost always noisier
Low dose imaging is almost always noisier
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Image Quality Noise
Signals in NoiseSignal is f
Noise is N
Signal-to-noise ratio
SNRa =amplitude(f )
amplitude(N)
SNRp =power(f )
power(N)
Hint: Units must match
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 72 / 412
Image Quality Noise
More on Signal-to-noiseSNR in decibels
SNR(dB) = 20 log10 SNRa
SNR(dB) = 10 log10 SNRp
Common example of SNR▶ signal height is A▶ noise standard deviation is �N▶ SNR is then
SNRa =A
�N
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 73 / 412
Image Quality Noise
Noise and Blurring Degrade Quality
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 74 / 412
Image Quality Artifacts
Nonrandom EffectsArtifacts: image features that do not correspond toa real object, and are not due to noise
▶ star artifact, beam hardening artifact▶ ring artifact, ghosts
Distortion: geometric or intensity changes notcorresponding to the real object
▶ magnification▶ barrel or pincushion distortion▶ quantization, saturation
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 75 / 412
Image Quality Artifacts
Common Artifacts
(a) motion(b) star(c) hardening(d) ring
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 76 / 412
Image Quality Accuracy
AccuracyAccuracy:
▶ conformity to truth→ quantitative accuracy
▶ clinical utility→ diagnostic accuracy
Quantitative accuracy:▶ numerical accuracy: bias, precision▶ geometric accuracy: dimensions
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 77 / 412
Image Quality Accuracy
Diagnostic QualityContingency table:
Disease
+ −
+ a b
Test
− c d
Variables:
a = # w/ disease & test says disease
b = # w/o disease & test says disease
c = # w/ disease & test says normal
d = # w/o disease & test says normal
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 78 / 412
Image Quality Accuracy
Diagnostic Accuracy
sensitivity =a
a + c
specificity =d
b + d
diagnostic accuracy =a + d
a + b + c + d
Disease
+ −
+ a b
Test
− c d
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 79 / 412
Image Quality Accuracy
Disease Prevalence
positive predictive value =a
a + b
negative predictive value =d
c + d
prevalence =a + c
a + b + c + d
Disease
+ −
+ a b
Test
− c d
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 80 / 412
Physics of Radiography
4 Physics of RadiographyX-ray ModalitiesAtomic StructureIonizing RadiationEnergetic ElectronsElectromagnetic RadiationEM StrengthEM AttenuationEM Dose
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 81 / 412
Physics of Radiography X-ray Modalities
X-ray ModalitiesChest x-rays
Mammography
Dental x-rays
Fluoroscopy
Angiography
Computed tomography
These do not involve radioactivity
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 82 / 412
Physics of Radiography Atomic Structure
Atomic Structurenucleons = {protons, neutrons}mass number A is # nucleons
atomic number Z is # protons
element symbol X is redundant with Z
nuclide is particular combination of nucleons▶
ZAX
▶ X -A
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 83 / 412
Physics of Radiography Atomic Structure
ElectronsOrbit in shells
Shell Number n Shell Label # Electrons 2n2
1 K ≤ 22 L ≤ 83 M ≤ 184 N ≤ 32
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 84 / 412
Physics of Radiography Atomic Structure
Electron Binding EnergyBasic principle:
bound energy < unbound energy + electron energy
Binding energy is difference
Binding energy of hydrogen electron: 13.6 eV
1 eV is the kinetic energy gained by an electron thatis accelerated across a one (1) volt potential
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 85 / 412
Physics of Radiography Atomic Structure
Ionization and ExcitationIonization is “knocking” an electron out of atom
▶ creates electron + ion
Excitation is “knocking” an electron to a higherorbit
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 86 / 412
Physics of Radiography Atomic Structure
Characteristic RadiationWhat happens to ionized or excited atom?
Return to ground state by rearrangement ofelectrons
Causes atom to give off energy
Energy given off as characteristic radiation▶ infrared▶ light▶ x-rays
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 87 / 412
Physics of Radiography Ionizing Radiation
Ionizing RadiationRadiation with energy > 13.6 eV is ionizing
Energy required to ionize:
▶ air ≈ 34 eV▶ lead ≈ 1 keV▶ tungsten ≈ 4 keV
These are average binding energies.
Radiation energies in medical imaging30 keV–511 keV
can ionize 10–40,000 atoms
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 88 / 412
Physics of Radiography Ionizing Radiation
Particulate RadiationConcerned with electron here (x-ray tube)(positron in later chapters)
Relativistic theory required (see text)
An electron accelerated across 100 kVpotential difference yields a 100 keV electron
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 89 / 412
Physics of Radiography Ionizing Radiation
Electromagnetic EM RadiationMany types of EM radiation:
▶ radio, microwaves,▶ infrared, visible light, ultraviolet▶ x-rays, gamma rays
electric and magnetic wave at right angles▶ waves with frequency �, or▶ “particles” (photons) with energy E
E = h–�
Planck’s constant h– = 4.14× 10−15 eV-sec
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 90 / 412
Physics of Radiography Energetic Electrons
Energetic Electron InteractionsTwo primary interactions:
▶ collisional transfer▶ radiative transfer
Collisional transfer:▶ Electron hits other electrons▶ Occasionally produces delta ray
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 91 / 412
Physics of Radiography Energetic Electrons
Energetic Electrons: Radiative TransferTwo types of radiative transfer:
▶ characteristic x-rays▶ bremsstrahlung x-rays
Characteristic x-rays:▶ electron ejects a K-shell electron▶ reorganization generates x-ray
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 92 / 412
Physics of Radiography Energetic Electrons
Energetic Electrons: BremsstrahlungBremsstrahlung x-rays
▶ Electron “grazes” nucleus, slows down▶ Energy loss generates x-ray
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 93 / 412
Physics of Radiography Energetic Electrons
X-ray Spectrum
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 94 / 412
Physics of Radiography Electromagnetic Radiation
EM InteractionsTwo important interactions:
▶ Photoelectric effect▶ Compton scattering
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 95 / 412
Physics of Radiography Electromagnetic Radiation
Photoelectric effectAtom completely absorbs incident photon
All energy is transferred
Atom produces▶ characteristic radiation, and/or▶ energetic electron(s)
Characteristic radiation might be▶ x-ray, or▶ light ← very important
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 96 / 412
Physics of Radiography Electromagnetic Radiation
Illustration of Photoelectric Effect
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 97 / 412
Physics of Radiography Electromagnetic Radiation
Compton ScatteringPhoton collides with outer-shell electron
Photon is deflected, angle �
Deflected photon has lower energy:
E ′ =E
1 + E (1− cos �)/(m0c2)
m0 is rest mass of electron
m0c2 = 511 keV
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 98 / 412
Physics of Radiography Electromagnetic Radiation
Illustration of Compton Scattering
When E higher▶ more Compton events scatter forward▶ Compton more of a problem
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 99 / 412
Physics of Radiography Electromagnetic Radiation
Probability of EM InteractionsPhotoelectric effect:
Prob[photoelectric event] ∝Z 4
eff(h–�)3
Photons are more penetrating at higherfrequencies/energies
Compton scattering:
Prob[Compton event] ∝ ED
ED approximately constant over diagnostic range
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 100 / 412
Physics of Radiography EM Strength
Beam Strength: Photon CountsPhoton fluence:
Φ =N
APhoton fluence rate:
� =N
AΔt
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 101 / 412
Physics of Radiography EM Strength
Beam Strength: Energy FlowEnergy fluence:
Ψ =Nh–�
AEnergy fluence rate:
=Nh–�
AΔt
Intensity: (= )
I (E ) =NE
AΔt
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 102 / 412
Physics of Radiography EM Strength
Polyenergetic Beam StrengthX-ray spectrum S(E ):
▶ S(E ) is the number of photons per unit energyper unit area per unit time
Photon fluence rate from spectrum:
� =
∫ ∞0
S(E ′) dE ′
Intensity from spectrum:
I =
∫ ∞0
E ′S(E ′) dE ′
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 103 / 412
Physics of Radiography EM Attenuation
EM Attenuation Geometries
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 104 / 412
Physics of Radiography EM Attenuation
“Good Geometry”, MonoenergeticNon-homogeneous slab:
dN
N= −�(x)dx
Integration yields:
N(x) = N0 exp{−∫ x
0
�(x ′)dx ′}
For intensity:
I (x) = I0 exp{−∫ x
0
�(x ′)dx ′}
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 105 / 412
Physics of Radiography EM Attenuation
Homogeneous SlabHomogeneous slab thickness Δx
Fundamental photon attenuation law
N = N0e−�Δx
� is linear attenuation coefficient
In terms of intensity:
I = I0e−�Δx
This is known as Beer’s Law
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 106 / 412
Physics of Radiography EM Attenuation
Half-value LayerHomogeneous slab (shielding)
HVL = thickness that willstop half the photons
1
2= exp{−� HVL}
Relation to �
HVL =0.693
�
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 107 / 412
Physics of Radiography EM Attenuation
“Good Geometry”, PolyenergeticMust deal with x-ray spectrum S0(E )
Abandon photon counting: use intensityFor heterogeneous materials
I (x) =
∫ ∞0
S0(E ′)E ′ exp
{−∫ x
0
�(x ′;E ′)dx ′}dE ′
Not very useful
Better to define effective energy,use monoenergetic approximation
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 108 / 412
Physics of Radiography EM Attenuation
Mass Attenuation Coefficientmass attenuation coefficient �/�
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 109 / 412
Physics of Radiography EM Dose
EM Radiation DoseHow many photons? → fluence
How much energy? → energy fluence
What does radiation do to matter?→ dose
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 110 / 412
Physics of Radiography EM Dose
Exposure: (the creation of ions)How many ions are created?
Exposure X , the number of ion pairs produced in aspecific volume of air by EM radiation
SI Units: C/kg
Common Units: roentgen, R
1 C/kg = 3876 R
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 111 / 412
Physics of Radiography EM Dose
Dose: (the deposition of energy)How much energy is deposited into material?
Dose, D, the energy deposited per unit volume
SI unit: Gray (Gy) 1 Gy = 1 J/kg
Common unit: rad
1 Gy = 100 rads
When X = 1 R soft tissue incurs 1 rad absorbeddose.
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 112 / 412
Physics of Radiography EM Dose
KermaHow much energy is deposited intothe electrons?
Kerma, K , is the energy deposited into the electronsof a material
SI units: Gray (Gy) = 1 J/kg = 100 rads
At diagnostic energies in the body, K = D
(In general, K ≥ D. Some electrons can causebremsstrahlung and their energy irradiated away →no dose. Not likely in body.)
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 113 / 412
Projection Radiography
5 Projection RadiographyRadiographic SystemsX-ray TubesFiltration, Restriction, and Contrast AgentsScatter ControlScreen and CassetteImaging EquationFilmNoise
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 114 / 412
Projection Radiography Radiographic Systems
Projection RadiographySystems:
▶ chest x-rays,mammography
▶ dental x-rays▶ fluoroscopy, angiography
Properties▶ high resolution▶ low dose▶ broad coverage▶ short exposure time
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 115 / 412
Projection Radiography Radiographic Systems
Radiographic System
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 116 / 412
Projection Radiography X-ray Tubes
X-ray Tube Diagram
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 117 / 412
Projection Radiography X-ray Tubes
X-ray Tube Components
Filament controls tube current (mA)
Cathode and focussing cup
Anode is switched to high potential▶ 30–150 kVp▶ Made of tungsten▶ Bremsstrahlung is 1%▶ Heat is 99%▶ Spins at 3,200–3,600 rpm
Glass housing; vacuum
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 118 / 412
Projection Radiography X-ray Tubes
Exposure ControlkVp applied for short duration
▶ fixed timer (SCR), or▶ automatic exposure control (AEC), 5 mm thick
ionization chamber triggers SCR
Tube current mA controlled by▶ filament current, and▶ kVp
mA times exposure time yields mAs
mAs measures x-ray exposure
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 119 / 412
Projection Radiography X-ray Tubes
X-ray Spectra
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 120 / 412
Projection Radiography Filtration, Restriction, and Contrast Agents
FiltrationInherent filtration
▶ Within anode▶ Glass housing
Added filtration▶ Aluminum▶ Copper/Aluminum
Note: Cu has 8keV characteristic xrays▶ Measured in mm Al/Eq
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 121 / 412
Projection Radiography Filtration, Restriction, and Contrast Agents
RestrictionGoal: To direct beam toward desired anatomy
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 122 / 412
Projection Radiography Filtration, Restriction, and Contrast Agents
Compensation FiltersGoal: to even out film exposure
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 123 / 412
Projection Radiography Filtration, Restriction, and Contrast Agents
Contrast AgentsGoal: To create contrast where otherwise none
10 20 30 40 50 60 70 80 100 150 200150.1
1.0
10
100Li
ne
ar A
tte
nua
tion C
oe
ffic
ient (c
m)
-1
Photon Energy (keV)
Hypaque
Kedge
37.4muscle
soft tissue
Bone
Fat
Kedge
33.2
BaSO
mix4
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 124 / 412
Projection Radiography Scatter Control
Scatter ControlIdeal x-ray path: a line!
Compton scattering causes blurring
How to reduce scatter?▶ airgap▶ scanning slit▶ grid
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 125 / 412
Projection Radiography Scatter Control
Grids
Effectiveness in scatter reduction?
grid ratio =h
b
6:1 to 16:1 (radiography) or 2:1 (mammo)
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 126 / 412
Projection Radiography Scatter Control
Problems with GridsRadiation is absorbed by grid
▶ grid conversion factor
GCF =mAs w/ grid
mAs w/o grid
▶ Typical range 3 < GCF < 8
Grid visible on x-ray film▶ move grid during exposure▶ linear or circular motion
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 127 / 412
Projection Radiography Screen and Cassette
Intensifying ScreenFilm stops only 1–2% of x-rays
Film stops light really well
Phosphor = calcium tungstate
Flash of light lasts 1× 10−10 second
∼1,000 light photons per 50 keV x-ray photon
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 128 / 412
Projection Radiography Screen and Cassette
Radiographic Cassette
Cassette holds two screens; makes “sandwich”
One side is leaded
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 129 / 412
Projection Radiography Imaging Equation
Basic Imaging Equation
I (x , y) =
∫ ∞0
S0(E ′)E ′ exp
{−∫ r(x ,y)
0
�(s;E ′, x , y)ds
}dE ′
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 130 / 412
Projection Radiography Imaging Equation
Geometric EffectsX-rays are diverging from source
Undesirable effects:
▶ cos3 � falloff across detector▶ anode heel effect▶ pathlength irregularities▶ magnification
I0 is intensity at (0, 0)
r is distance from (x , y) to x-ray origin
� is angle between (0, 0) and (x , y)
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 131 / 412
Projection Radiography Imaging Equation
Inverse Square LawNet flux of photons decrease as 1/r 2.Therefore
I0 =IS
4�d2Ir =
IS4�r 2
Eliminate source intensity IS
Ir = I0d2
r 2
Since cos � = d/r
Ir = I0 cos2 �
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 132 / 412
Projection Radiography Imaging Equation
Obliquity
Intensity isId = I0 cos �
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 133 / 412
Projection Radiography Imaging Equation
Beam Divergence and Flat DetectorInverse square law and obliquity combine
Id(xd , yd) = I0 cos3 �
Can usually be ignored. Why?
▶ Detector is far away▶ Field of view (FOV) is often small
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 134 / 412
Projection Radiography Imaging Equation
Anode Heel EffectIntensity within the x-ray cone
▶ Not uniform▶ stronger in the cathode direction▶ 45% variation is typical
Compensate, use to advantage, or ignore
We will ignore in math
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 135 / 412
Projection Radiography Imaging Equation
Path Length of SlabUniform slab yields different intensities
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 136 / 412
Projection Radiography Imaging Equation
Effect of Pathlength on IntensityIntensity on detector
Id(x , y) = I0 exp{−�L/ cos �}
Including inverse square law and obliquity:
Id(x , y) = Ii cos3 � exp{−�L/ cos �}
If d ≈ r all effects can be ignored
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 137 / 412
Projection Radiography Imaging Equation
Object MagnificationSize on detector depends on distance from source
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 138 / 412
Projection Radiography Imaging Equation
Magnification FormulaObject at position z from source
Height of object is w .
Height wz on detector is
wz = wd
z
Magnification is
M(z) =d
zCan lead to edge blurring and misleading sizes
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 139 / 412
Projection Radiography Imaging Equation
Thin Slab Imaging EquationThin slab at z of �(x , y)Let “transmittivity” be
tz(x , y) = exp{−�(x , y)Δz}
On detector, intensity is
Id(x , y) = I0 cos3 � tz
(x
M(z),
y
M(z)
)After substitution
Id(x , y) = I0
(d√
d2 + x2 + y 2
)3
tz(xzd,yz
d
)Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 140 / 412
Projection Radiography Imaging Equation
Sources of BlurringExtended source
Intensifier screen
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 141 / 412
Projection Radiography Imaging Equation
Extended Source
z
d
ExtendedX-raySource
Image ofExtendedSource
PointHole
DetectorPlane
s x y( , )D
D´
Source spatial distribution: s(x , y)
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 142 / 412
Projection Radiography Imaging Equation
Source MagnificationSource diameter on detector:
D ′ =d − z
zD
Source magnification:
m(z) = −d − z
z= 1−M(z)
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 143 / 412
Projection Radiography Imaging Equation
Source BlurringImage of source through pinhole at z
Id(x , y) =1
4�d2m2s( xm,y
m
)Intensity at detector:
Id(x , y) =cos3 �
4�d2m2tz( x
M,y
M
)∗ s( xm,y
m
)
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 144 / 412
Projection Radiography Imaging Equation
Film-Screen BlurringFilm
X-rayPhoton
r
x
L
Light Photons
Phosphors
Film-screen impulse response: h(x , y)
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 145 / 412
Projection Radiography Imaging Equation
Overall Imaging EquationInclude all geometric effects
Id(x , y) = cos3 �1
4�d2m2s( xm,y
m
)∗ tz
( x
M,y
M
)∗ h(x , y)
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 146 / 412
Projection Radiography Film
FilmDeveloped film
Optical transmissivity
T =ItIi
Optical density
D = log10
IiIt
Note: O = 1/T is optical opacity
Usable densities 0.25 < D < 2.25
Best densities 1.0 < D < 1.5
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 147 / 412
Projection Radiography Film
H & D CurveOptical density from x-ray exposurefor film-screen combination:
10-1
100
101
102
103
0
0.5
1
1.5
2
2.5
3
3.5
4
(a) High Speed
Film with
CaWO
Screens4
(b) Direct X-ray
Film
(c) High Speed
Film Without
Screens
Fog Level
Exposure, mR
Op
tica
l De
nsi
ty
Toe
Linear
Region
Shoulder
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 148 / 412
Projection Radiography Film
X-ray Exposure to Film DensityX-ray exposure yields optical density
D = Γ log10
X
X0
Γ is film gamma
Typical ranges: 0.5 < Γ < 3.0
Latitude is range exposures where relationship islinear
Speed is inverse of exposure at which
D = 1 + fog level
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 149 / 412
Projection Radiography Noise
NoiseLocal contrast
C =It − IbIb
Signal is It − IbNoise is due to Poisson behavior
Variance of noise in background: �2b
Signal to noise
SNR =It − Ib�b
=CIb�b
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 150 / 412
Projection Radiography Noise
Signal-to-noiseModel x-ray burst as monoenergetic
▶ effective energy is h�▶ background intensity is
Ib =Nbh�
AΔt
Signal-to-noise is
SNR = C√
Nb
More photons is better
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 151 / 412
Projection Radiography Noise
Detective Quantum EfficiencyHow good is a detector?
Consider:▶ Potential SNR before detection▶ Actual SNR upon detection
Detective Quantum Efficiency
DQE =
(SNRoutSNRin
)2
Degradation of SNR during detection
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 152 / 412
Projection Radiography Noise
Compton ScatterCompton adds intensity “fog”: IsResulting contrast
C ′ =C
1 + Is/Ib
Resulting SNR
SNR′ =SNR√
1 + Is/Ib
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 153 / 412
Computed Tomography
6 Computed TomographyOverviewCT GenerationsSystem ComponentsCT MeasurementsRadon TransformReconstructionProjection-Slice TheoremResolutionNoiseFan Beam Reconstruction
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 154 / 412
Computed Tomography Overview
Computed TomographyTomography:
▶ image of slice▶ removes “overlaying structure”▶ improves contrast within slice
Computed:▶ requires computer▶ reconstruction algorithm
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 155 / 412
Computed Tomography Overview
1-D Projection“fan beam” collimation
row of electronic detectors
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 156 / 412
Computed Tomography Overview
Premise of CTA single 1-D projection is not informative
Many 1-D projections▶ permit slice reconstruction▶ many angular views is the key
http://www.gehealthcare.com
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 157 / 412
Computed Tomography CT Generations
1G CT Scanner
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 158 / 412
Computed Tomography CT Generations
2G CT Scanner
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 159 / 412
Computed Tomography CT Generations
3G CT Scanner
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 160 / 412
Computed Tomography CT Generations
4G CT Scanner
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 161 / 412
Computed Tomography CT Generations
Electron Beam (5G) CT Scanner
http://radiology.rsnajnls.org
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 162 / 412
Computed Tomography CT Generations
Gantry, Slipring, and Table
http://www.gehealthcare.com http://www.cissincorp.com
1–2 revolutions per secondJerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 163 / 412
Computed Tomography CT Generations
Helical (6G) CT ScannerStep-wise table movement yields stack of 2D slices
Continuous table movement yields stream of 1Dprojections
3D volume is reconstructed from helical acquisition
http://imaging.cancer.gov
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 164 / 412
Computed Tomography CT Generations
Multi-slice (7G) CT ScannerFeatures:
▶ 16–64 parallel detectorrows
▶ 14,336–57,344 detectorelements
▶ 20–80 mm detector“height”
▶ 16–64 0.5mm slices witheach second gantryrevolution
CT is becoming “cone beam”
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 165 / 412
Computed Tomography System Components
X-ray Tubes in CTUse only one tube
▶ exception: EBCT▶ exception: dual-source CT
80kVp–140kVp, continuous excitation▶ dual-energy is possible
fan-beam (1–10 mm thick), or
thin-cone collimation 20–80 mm
More filtering than projection radiography▶ copper followed by aluminum▶ Better approximation to monoenergetic
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 166 / 412
Computed Tomography System Components
CT DetectorsMost are solid-state:
▶ scintillation crystal▶ solid state photo-diode
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 167 / 412
Computed Tomography System Components
CT Detector SpecificationsSingle-slice scanners:
▶ Area: 1.0 mm × 15.0 mm▶ Thick in 3G, thin in 4G & EBCT
Multi-slice scanners:▶ Area: 1.0 mm × 1.25 mm▶ Grouped in multiples of 1.25 mm
Xenon gas detectors for less expensive scanners
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 168 / 412
Computed Tomography CT Measurements
CT Measurement ModelMonoenergetic model:
Id = I0 exp
{−∫ d
0
�(s; E )ds
}E is effective energy
E is that energy which in a given materialwill produce the same measured intensityfrom a monoenergetic source as from theactual polyenergetic source.
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 169 / 412
Computed Tomography CT Measurements
CT MeasurementObserve IdRearrange monoenergetic model:
gd = − lnIdI0
=
∫ d
0
�(s; E )ds
gd is a line integral of the linear attenuationcoefficient at the effective energy
Note: Requires calibration measurement of I0
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 170 / 412
Computed Tomography CT Measurements
CT NumbersConsistency across CT scanners desired
CT number is defined as:
h = 1000× �− �water�water
h has Hounsfield units (HU)
Usually rounded or truncated to nearestinteger
Range: −1,000 to ∼3,000
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 171 / 412
Computed Tomography Radon Transform
Describing LinesPossible descriptions of lines:
▶ Functional: y = ax + b▶ Parametric: (x(s), y(s))▶ Set: {(x , y)∣(x , y) are on a line}
Critique:▶ Functional: what about vertical lines???▶ Parametric: good for model of process▶ Set: good for theory of reconstruction
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 172 / 412
Computed Tomography Radon Transform
Picture of a Line
l
l
L( , )l θ
θ
f(x,y)
x
y
0
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 173 / 412
Computed Tomography Radon Transform
Line ParametersDescribed by:
▶ Orientation or angle, �▶ Lateral translation or position, ℓ
Written as L(ℓ, �)
L(ℓ, �) = {(x , y)∣(x , y) are on the line
with position ℓ
and angle �}
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 174 / 412
Computed Tomography Radon Transform
Line Integral: parametric formWhat is integral of f (x , y) on L(ℓ, �)?
Step 1: Parameterize L(ℓ, �):
x(s) = ℓ cos � − s sin �
y(s) = ℓ sin � + s cos �
Step 2: Integrate f (x , y) over parameter s
g(ℓ, �) =
∫ ∞−∞
f (x(s), y(s))ds
Use this form for the forward problem
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 175 / 412
Computed Tomography Radon Transform
Line Integral: set formIntegrate over whole plane;non-zero only on L(ℓ, �)
Key is sifting property
q(ℓ) =
∫ ∞−∞
q(ℓ′)�(ℓ′ − ℓ)dℓ′
Use line impulse on L(ℓ, �)
g(ℓ, �) =∫ ∞−∞
∫ ∞−∞
f (x , y)�(x cos � + y sin � − ℓ) dxdy
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 176 / 412
Computed Tomography Radon Transform
Physical Meanings of f (x , y) and g(ℓ, �)Recall monoenergetic model:
Id = I0 exp
{−∫ d
0
�(s; E )ds
}Rearrange:
− lnIdI0
=
∫ d
0
�(x(s), y(s); E )ds
Relationship is:
f (x , y) = �(x , y ; E )
g(ℓ, �) = − lnIdI0
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 177 / 412
Computed Tomography Radon Transform
What is g(ℓ, �)?Fix ℓ and �: line integral of f (x , y)
Fix �: projection of f (x , y) at angle �
Function of � and ℓ:g(ℓ, �) is the Radon transform of f (x , y)
g(ℓ, �) = ℛ{f (x , y)}
Transform . . . hmmm . . . can we find aninverse transform?
f (x , y) = ℛ−1{g(ℓ, �)}
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 178 / 412
Computed Tomography Radon Transform
SinogramCT data acquired for collection of ℓ and �
CT scanners acquires a sinogram
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 179 / 412
Computed Tomography Reconstruction
BackprojectionGoal: find f (x , y) from g(ℓ, �)Strategy: “smear” g(ℓ, �) into planeFormally:
b�(x , y) = g(x cos � + y sin �, �)
b�(x , y) is a laminar image
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 180 / 412
Computed Tomography Reconstruction
Backprojection Summation“Add up” all the backprojection images:
fb(x , y) =
∫ �
0
b�(x , y)d�
=
∫ �
0
g(x cos � + y sin �, �)d�
=
∫ �
0
[g(ℓ, �)]ℓ=x cos �+y sin � d�
fb(x , y) is called a laminogram orbackprojection summation image
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 181 / 412
Computed Tomography Reconstruction
Properties of Laminogram“Bright spots” tend to reinforce
Problem:fb(x , y) ∕= f (x , y)
What is wrong?
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 182 / 412
Computed Tomography Reconstruction
Convolution BackprojectionCorrect reconstruction formula:
f (x , y) =
∫ �
0
[c(ℓ) ∗ g(ℓ, �)]ℓ=x cos �+y sin � d�
wherec(ℓ) = ℱ−1{∣%∣}
is called the ramp filter.
Three steps: ← know/understand these!!▶ 1. convolution▶ 2. backprojection▶ 3. summation
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 183 / 412
Computed Tomography Reconstruction
Step 1: ConvolutionConvolve every projection with c(ℓ)
the horizontal direction in a sinogram
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 184 / 412
Computed Tomography Reconstruction
Step 2: Backprojection1D projection → 2D laminar function
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 185 / 412
Computed Tomography Reconstruction
Step 3: SummationAccumulate sum of backprojection images
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 186 / 412
Computed Tomography Projection-Slice Theorem
Projection-Slice TheoremRadon transform:
g(ℓ, �) = ℛ{f (x , y)}
Fourier transforms:
G (%, �) = ℱ1D {g(ℓ, �)}F (u, v) = ℱ2D {f (x , y)}
Projection-slice theorem:
G (%, �) = F (% cos �, % sin �)
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 187 / 412
Computed Tomography Projection-Slice Theorem
Illustration of Projection-Slice Theorem
x
y
u
v
ρ
l
θ
θ
f(x,y) F(u,v)
2D Fourier Transform
1D Fourier Transform
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 188 / 412
Computed Tomography Projection-Slice Theorem
Exact Reconstruction FormulasFourier reconstruction:
f (x , y) = ℱ−1
2D {G (%, �)}
Filtered backprojection:
f (x , y) =
∫ �
0
[∫ ∞−∞∣%∣G (%, �)e+j2�%ℓd%
]ℓ=x cos �+y sin �
d�
Convolution backprojection:
f (x , y) =
∫ �
0
∫ ∞−∞
g(ℓ, �)c(x cos � + y sin � − ℓ)dℓd�
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 189 / 412
Computed Tomography Projection-Slice Theorem
Ramp Filter Design∣%∣ is not integrable
⇒ c(ℓ) does not exist
Actual ramp filter is designed as
c(ℓ) = ℱ−1
1D{W (%)∣%∣}
Simplest window function is
W (%) = rect
(%
2%0
)
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 190 / 412
Computed Tomography Resolution
Factors Affecting CT ResolutionDetector width ∼ area detectors
detector indicator function = s(ℓ)
Window function W (%)Approximate CBP:
f (x , y) =∫ �
0
[∫ ∞−∞
G (%, �)S(%)W (%)∣%∣e j2�%ℓ d%]ℓ=x cos �+y sin �
d�
whereS(%) = ℱ{s(ℓ)}
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 191 / 412
Computed Tomography Resolution
Blurry ReconstructionBlurry projection:
g(ℓ, �) = g(ℓ, �) ∗ s(ℓ) ∗ w(ℓ)
= g(ℓ, �) ∗ h(ℓ)
Radon transform convolution theorem
ℛ{f ∗2 h} = ℛ{f } ∗1 ℛ{h}
Leads to
f (x , y) = f (x , y) ∗ ℛ−1{h(ℓ)}
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 192 / 412
Computed Tomography Resolution
Circular Symmetry of BlurringCT image blurred by convolution kernel
h(x , y) = ℛ−1{h(ℓ)}
Fourier transform of h(ℓ)
H(%) = ℱ1{h(ℓ)} = S(%)W (%)
which is independent of �.
Therefore, H(u, v) is circularly symmetric
H(q) = ℱ2{h(x , y)} = S(q)W (q)
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 193 / 412
Computed Tomography Resolution
PSF Given by Hankel TransformPSF is circularly symmetric and given by
h(r) = ℋ−1{S(%)W (%)}
Reconstructed image given by
f (x , y) = f (x , y) ∗ h(r)
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 194 / 412
Computed Tomography Noise
Noise in CT MeasurementsBasic measurement is:
gij = − ln
(Nij
N0
)
▶ line Lij▶ angle i▶ position j
Noise is “in” Poisson random variable Nij
▶ mean Nij
▶ variance Nij
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 195 / 412
Computed Tomography Noise
Functions of Random VariablesIt follows that gij is a random variable
gij ≈ ln
(N0
Nij
)Var(gij) ≈
1
Nij
�(x , y) is approximate reconstruction
It follows that �(x , y) is a random variable
What are the mean and variance of �?
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 196 / 412
Computed Tomography Noise
CBP ApproximationConvolution backprojection (CBP):
�(x , y) =
∫ �
0
∫ ∞−∞
g(ℓ, �)c(x cos � + y sin � − ℓ)dℓd�
Approximations:▶ M angles; Δ� = �/M▶ N + 1 detectors; Δℓ = T▶ c(ℓ) ≈ c(ℓ)
Discrete CBP:
�(x , y) =( �M
) M∑j=1
T
N/2∑i=−N/2
g(iT , j�/M)c(x cos �j+y sin �j−iT )
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 197 / 412
Computed Tomography Noise
More Definitions and ApproximationsNij is mean for i-th detector and j-th angle
Nij is independent for different measurements
Nij = N , an “object uniformity” assumption
c(ℓ) is created using rectangular window W (%) withcutoff %0.
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 198 / 412
Computed Tomography Noise
ConclusionsMean(�) is desired result
Var(�) = �2� is inaccuracy
�2� ≈
2�2
3%3
0
1
M
1
N/T
Be cautious on conclusions: not all variables areindependent in a real physical system
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 199 / 412
Computed Tomography Noise
Signal-to-noise RatioDefinition (usual)
SNR =C �
��
After substitution:
SNR =C �
�
√3M
2%30
N
T
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 200 / 412
Computed Tomography Noise
SNR in a Good DesignWhat should %0 be?
Let dectector width = w
%0 should be anti-aliasing filter:
%0 =k
wwhere k ≈ 1
In 3G scanner w = T
ThenSNR ≈ 0.4kC �w
√NM
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 201 / 412
Computed Tomography Noise
SNR in Fan-Beam CaseDefinitions:
▶ Nf is mean photon count per fan▶ D is number of detectors▶ L is length of detector array
Then
SNR ≈ 0.4kC �L
√NfM
D3
Strange: In 3G, increasing D decreases SNR.
Reason: This analysis ignores resolution
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 202 / 412
Computed Tomography Noise
Rule of ThumbVariables:
▶ D is number of detectors▶ M is number of angles▶ J2 is number of pixels in image
Very approximate “rule”:
D ≈ M ≈ J
Typical numbers:
Lo: D ≈ 700 M ≈ 1, 000 J ≈ 512
Hi: D ≈ 900 M ≈ 1, 600 J ≈ 1, 024
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 203 / 412
Computed Tomography Fan Beam Reconstruction
Fan Beam Geometry
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 204 / 412
Computed Tomography Fan Beam Reconstruction
Sinogram Rebinning
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 205 / 412
Computed Tomography Fan Beam Reconstruction
Fan-Beam Variables
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 206 / 412
Computed Tomography Fan Beam Reconstruction
Fan-Beam Convolution BackprojectionFormula
f (x , y) =
∫ 2�
0
1
(D ′)2
∫ m
− mp( , �)c ′( ′ − )d d�
D ′ depends on (x , y)
c ′ is a different filter than c
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 207 / 412
Physics of Nuclear Medicine
7 Physics of Nuclear MedicineBinding EnergyRadioactivityRadiotracers
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 208 / 412
Physics of Nuclear Medicine Binding Energy
NomenclatureAtomic number: Z , number of protons in nucleus
Mass number: A, number of nucleons in nucleus
Nuclide: unique combination of protons andneutrons in nucleus
Radionuclide: a nuclide that is radioactive
Isotope: atoms with same Z , different A
Isobar: atoms with same A, different Z
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 209 / 412
Physics of Nuclear Medicine Binding Energy
Mass Defect and Binding EnergyMass defect =
Mass of constituents of atom
− actual mass of atom
unified mass unit, u, = 1/12 mass of C-12 atom
Binding energy = mass defect ×c2
One u is equivalent to 931 MeV
Generally, more massive atom, more binding energy
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 210 / 412
Physics of Nuclear Medicine Binding Energy
Binding Energy per Nucleon
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 211 / 412
Physics of Nuclear Medicine Radioactivity
What is Radioactivity?Radioactive decay: rearrangement of nucleii tolower energy states = greater mass defect
Parent atom decays to daughter atom
Daughter has higher binding energy/nucleon thanparent
A radioatom is said to decay when its nucleus isrearranged
A disintegration is a radioatom undergoingradioactive decay.
Energy is released with disintegration.
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 212 / 412
Physics of Nuclear Medicine Radioactivity
“Line” of StabilityNuclides divide into two groups:
▶ Non-radioactive — i.e., stable atoms▶ Radioactive — i.e., unstable atoms
“Line” of stability:
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 213 / 412
Physics of Nuclear Medicine Radioactivity
Decay ModesFour main modes of decay:
▶ alpha particles (2 protons, 2 neutrons)▶ beta particles (electrons)▶ positrons (anti-matter electrons)▶ isomeric transition (gamma rays produced)
Medical imaging is only concerned with:▶ positrons (PET), and▶ gamma rays (scintigraphy, SPECT)
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 214 / 412
Physics of Nuclear Medicine Radioactivity
Measurement of RadioactivityRadioactivity, A, # disintegrations per second
1 Bq = 1 dps
1 Ci = 3.7× 1010 Bq
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 215 / 412
Physics of Nuclear Medicine Radioactivity
Radioactive Decay LawTime evolution of radioactivity:
At = A0e−�t
� is the decay constant
Half-life t1/2 is defined by
At1/2
A0=
1
2= e−�t1/2
It follows that
t1/2 =0.693
�
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 216 / 412
Physics of Nuclear Medicine Radioactivity
Statistics of DecayOver “short” time Δt relative to t1/2:
# radioatoms N0 approximately constant
Statistics are Poisson:
P[ΔN = k] =(�N0Δt)ke−�N0Δt
k!
Interpretation: �N0Δt is probability of having onedisintegration from N0 radioatoms in time intervalΔt
�N0 is called Poisson rate, units aredisintegrations per second, it is activity
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 217 / 412
Physics of Nuclear Medicine Radiotracers
RadiotracersRadionuclides in the body must be safe
By themselves:▶ Iodine-123,▶ Iodine-131
Labeled: Chemically attached to natural substances:
▶ Technetium-99m labeled DTPA,▶ Oxygen-15 labeled O2,▶ Fluorine-18 labeled glucose
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 218 / 412
Physics of Nuclear Medicine Radiotracers
Radiotracer PropertiesEmit gamma rays or positrons
Half life: minutes to a few hours
Positron emission:
▶ positrons annihilate▶ produces two 511 keV gamma rays,▶ gamma rays are 180-degrees apart
Gamma ray emission:
▶ monoenergetic gamma rays (desirable)▶ high energy gamma rays (desirable)
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 219 / 412
Physics of Nuclear Medicine Radiotracers
Some RadiotracersGamma Ray Emitters:
▶ Iodine-123 (13.3 h, 159 keV)▶ Iodine-131 (8.04 d, 364 keV)▶ Iodine-125 (60 d, 35 keV) (Bad. Why?)▶ Thallium-201 (73 h, 135 keV)▶ Technetium-99m (6 h, 140 keV)
Positron Emitters:▶ Fluorine-18 (110 min, 202 keV)▶ Oxygen-15 (2 min, 696 keV)
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 220 / 412
Planar Scintigraphy
8 Planar ScintigraphyScintigraphy SystemsGamma CameraAcquisition ModesImage EquationResolution and SensitivityArtifacts and Noise
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 221 / 412
Planar Scintigraphy Scintigraphy Systems
Broad PurposeGamma emitter in body; where is it?
Planar camera; like radiography
2D projection of 3D concentration
X-ray Image Bone Scintigram
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 222 / 412
Planar Scintigraphy Scintigraphy Systems
A SPECT/Scintigraphy/CT System
http://www.gehealthcare.com
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 223 / 412
Planar Scintigraphy Scintigraphy Systems
Example PreparationGallium-67 citrate
half-life is 78 hr
93 keV (40%), 184 keV (24%), 296 keV (22%), and388 keV (7%).
150-220 MBq (4-6 mCi) intravenously
48 hr after injection, about 75% remains in body
equally distributed among the liver, bone and bonemarrow, and soft tissues.
Scintigrams 24–72 hrs after injection
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 224 / 412
Planar Scintigraphy Scintigraphy Systems
Whole Body Image
http://www.gehealthcare.com
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 225 / 412
Planar Scintigraphy Gamma Camera
Gamma/Anger Camera Components
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 226 / 412
Planar Scintigraphy Gamma Camera
Collimators
(a) (b)
(c)(d)
(a) Parallel hole
(b) Converging hole (magnifies)
(a) Diverging hole (minifies)
(a) Pin-hole (2–5 mm)
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 227 / 412
Planar Scintigraphy Gamma Camera
DetectorSingle large-area NaI(Tl) crystal
Diameters:▶ 30–50 cm in diameter▶ Mobile units: 30 cm▶ Fixed scanners: 50 cm
Thickness:▶ High-E emitters: 1.25 cm thick▶ Low-E emitters: 6–8 mm thick
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 228 / 412
Planar Scintigraphy Gamma Camera
Photomultiplier Tube Array
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 229 / 412
Planar Scintigraphy Gamma Camera
Photomultiplier Tube
Dynodes
Focussing
Grid
Photocathode
Light Photons
1,200 V
Output Signal
Anode
e-
e-
e-
e-
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 230 / 412
Planar Scintigraphy Gamma Camera
Pulse HeightResponse to single gamma ray photon
PMT responses, ak , k = 1, . . . ,K
Total response of cammera is Z -pulse
Z =K∑
k=1
ak
Height of Z pulse is important▶ Can remove Compton photons▶ Can reject multiple hits
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 231 / 412
Planar Scintigraphy Gamma Camera
Pulse Height Analysis
Discriminator circuit rejectsnon-photopeak events
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 232 / 412
Planar Scintigraphy Gamma Camera
Event Positioning LogicTube centers at (xk , yk) k = 1, . . . ,K
Center of mass of pulse responses is
X =1
Z
K∑k=1
xkak
Y =1
Z
K∑k=1
ykak
This is pulse location
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 233 / 412
Planar Scintigraphy Acquisition Modes
Acquisition ModesHow to use the camera to make images?
▶ List mode▶ Static frame mode▶ Dynamic frame mode▶ Multiple-gated acquisition▶ Whole body mode
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 234 / 412
Planar Scintigraphy Acquisition Modes
List Mode
Complete information, but memory hog
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 235 / 412
Planar Scintigraphy Acquisition Modes
Static Frame Mode
Matrix sizes: 64 × 64, 128 × 128, 256 × 256
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 236 / 412
Planar Scintigraphy Acquisition Modes
Dynamic Frame Mode
Useful for imaging transient physiological processes
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 237 / 412
Planar Scintigraphy Acquisition Modes
Multiple Gated Acquisition
Cardiac (ECG) gated. Data resorted using ECG
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 238 / 412
Planar Scintigraphy Acquisition Modes
Whole Body Mode
Common in bone scans and tumor screening
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 239 / 412
Planar Scintigraphy Image Equation
Imaging Geometry and Assumptions
Lines defined by (parallel) collimator holesIgnore Compton scatteringRadioactivity is A(x , y , z)Monoenergetic photons, energy E
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 240 / 412
Planar Scintigraphy Image Equation
Imaging EquationPhoton fluence on detector is
�(x , y) =∫ 0
−∞
A(x , y , z)
4�z2e−∫ 0
z
�(x , y , z ′;E )dz ′
dz
Depth-dependent effects from:▶ inverse square law, and▶ object-dependent attenuation
Consequences:▶ Near activity brighter▶ Front and back are different
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 241 / 412
Planar Scintigraphy Image Equation
Planar SourcesAz0
(x , y) has radioactivity on z = z0
A(x , y , z) = Az0(x , y)�(z − z0)
Detected photon fluence rate
�(x , y) = Az0(x , y)
1
4�z20
exp
{−∫ 0
z0
�(x , y , z ′;E )dz ′}
Two terms attenuate desired result▶ inverse square law: constant for (x , y)▶ �: not constant for (x , y)
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 242 / 412
Planar Scintigraphy Resolution and Sensitivity
Collimator Resolution
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 243 / 412
Planar Scintigraphy Resolution and Sensitivity
Collimator ResolutionCollimator Resolution = FWHM =
RC (∣z ∣) =d
l(l + b + ∣z ∣)
Gaussian approximation
hc(x , y ; ∣z ∣) = exp{−4(x2 + y 2) ln 2/R2
C (∣z ∣)}
Planar source is blurred
�(x , y) = Az0(x , y)
1
4�z20
×
exp
{−∫ 0
z0
�(x , y , z ′;E )dz ′}∗ hc(x , y ; ∣z0∣)
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 244 / 412
Planar Scintigraphy Resolution and Sensitivity
Instrinsic ResolutionWhere did the x-ray photon hit?
▶ Compton in crystal spreads out light▶ Crystal thickness▶ Noise in light, PMTs, and electronics
Gaussian approximation
hI (x , y) = exp{−4(x2 + y 2) ln 2/R2
I
}Planar source is further blurred
�(x , y) = Az0(x , y)
1
4�z20
exp
{−∫ 0
z0
�(x , y , z ′;E )dz ′}
∗hC (x , y ; ∣z0∣) ∗ hI (x , y)
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 245 / 412
Planar Scintigraphy Resolution and Sensitivity
Collimator SensitivityCollimator Efficiency = Sensistivity =
� =
(Kd2
l(d + h)
)2
where K ≈ 0.25.
� is the fraction of photons (on average) that passthrough the collimator for each emitted photondirected at the camera
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 246 / 412
Planar Scintigraphy Resolution and Sensitivity
Resolution vs. Sensitivity
Table: Resolution and Sensitivity for Several Collimators
collimator d (mm) l (mm) h (mm) resolution relative@ 10 cm (mm) sensitivity
LEUHR 1.5 38 0.20 5.4 12.1LEHR 1.9 38 0.20 6.9 20.5LEAP 1.9 32 0.20 7.8 28.9LEHS 2.3 32 0.20 9.5 43.7
LEUHR = low energy ultra-high resolutionLEHR = low energy high resolutionLEAP = low energy all purposeLEHS = low energy high sensitivity
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 247 / 412
Planar Scintigraphy Resolution and Sensitivity
Detector EfficiencyDepends on crystal thickness
▶ thicker ⇒ more efficient▶ 100% at 100keV; 10-20% at 511keV
Tradeoff:▶ If E low ⇒ use thinner crystal
★ better intrinsic resolution▶ If E high ⇒ use thicker crystal
★ poorer intrinsic resolution
▶ Higher E , less abosorption in body
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 248 / 412
Planar Scintigraphy Artifacts and Noise
Geometry and NonuniformityGeometric distortion
▶ pincushion distortion▶ barrel distortion▶ wavy line distortion
Image nonuniformity▶ variation as much as 10%▶ non-uniform detector efficiencies▶ geometric distortions → “hot spot”▶ edge packing
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 249 / 412
Planar Scintigraphy Artifacts and Noise
Image SNRSuppose N photons are detected
Then intrinsic SNR of frame mode is
SNR(intrinsic) =
√N
J
J2 is number of pixels in image
For similar areas of target and background:
SNR = C√Nb
Just like in projection radiography
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 250 / 412
Planar Scintigraphy Artifacts and Noise
Energy ResolutionEnergy resolution = FWHM of photopeak
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 251 / 412
Planar Scintigraphy Artifacts and Noise
Pulse PileupPulse pileup = two simultaneous -rays
Event rejected▶ because of energy discrimination▶ wasted photons
Cannot improve image using larger dose
Instead, keep dose low and image longer
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 252 / 412
Emission Tomography
9 Emission TomographyOverviewSPECT System ComponentsSPECT Imaging EquationSPECT ReconstructionPrinciple of PETPET System ComponentsPET Imaging EquationPET ReconstructionResolution and Artifacts
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 253 / 412
Emission Tomography Overview
OverviewSPECT
▶ uses gamma ray emitters▶ uses Anger camera▶ 3-D volume reconstruction
PET▶ uses positron emitters▶ requires coincidence detectors▶ multiple 2-D slices
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 254 / 412
Emission Tomography SPECT System Components
SPECT HardwareRotating gamma camera
Each “row” is separate slice
Multiple heads (2 or 3) are common
High-performance cameras used▶ < 1% nonuniformity required▶ need good mechanical alignment
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Emission Tomography SPECT System Components
Typical SPECT System
http://www.gehealthcare.com
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Emission Tomography SPECT System Components
Multiple Head Tradeoffs
Table: Comparison of acquisition times and relative sensitivities forsingle- and multi-head systems with identical camera heads andcollimation.
360∘ 180∘
Acq Time Rel Sens Acq Time Rel Sens
Single 30 1 30 1Double (heads@180∘) 15 2 30 1Double (heads@90∘) 15 2 15 2Triple 10 3 20 1.5
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Emission Tomography SPECT Imaging Equation
SPECT Coordinate System
“Home position:” x → z , y → ℓ, z → y
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Emission Tomography SPECT Imaging Equation
Basic Imaging EquationParallel hole collimators
Camera fixed distance R from origin(origin in patient)
Imaging equation in “home” position:
�(z , ℓ) =
∫ R
−∞
A(x , y , z)
4�(y − R)2
exp
{−∫ R
y
�(x , y ′, z ;E )dy ′}dy
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Emission Tomography SPECT Imaging Equation
Tomographic Imaging Geometry
z is irrelevant
Line described by
L(ℓ, �) = {(x , y)∣ x cos � + y sin � = ℓ}
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Emission Tomography SPECT Imaging Equation
Tomographic Imaging Equation
�(ℓ, �) =
∫ R
−∞
A(x(s), y(s))
4�(s − R)2
exp
{−∫ R
s
�(x(s ′), y(s ′);E )ds ′}ds
Two unknowns: A(x , y) and �(x , y)
Generally intractable ⇒▶ ignore attenuation (often done)▶ assume constant▶ measure and apply atten correction
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 261 / 412
Emission Tomography SPECT Imaging Equation
Approximate SPECT Imaging EquationBold approximations: ignore attenuation, inversesquare law, and scale factors:
�(ℓ, �) =
∫ ∞−∞
A(x(s), y(s))ds
Using line impulse:
�(ℓ, �) =∫ ∞−∞
∫ ∞−∞
A(x , y)�(x cos � + y sin � − ℓ) dx dy
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 262 / 412
Emission Tomography SPECT Reconstruction
SPECT ReconstructionRecognize:
f (x , y) = A(x , y)
g(ℓ, �) = �(ℓ, �)
Use convolution backprojection
A(x , y) =∫ �
0
∫ ∞−∞
�(ℓ, �)c(x cos � + y sin � − ℓ) dℓ d�
Approximate ramp filter:
c(ℓ) = ℱ−11D {∣%∣W (%)}
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 263 / 412
Emission Tomography Principle of PET
PET PrinciplesPositron emitters
Positron annihilation:▶ short distance from emission▶ produces two 511 keV gamma rays▶ gamma rays 180∘ opposite directions
Principle: detect coincident gamma rays
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 264 / 412
Emission Tomography Principle of PET
A PET Scanner
Used with permission of GE Healthcare
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Emission Tomography Principle of PET
Positron Annihilation
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Emission Tomography Principle of PET
Annihilation Coincidence Detection (ACD)Event occurs if detections are coincident
Time window is typically 2–20 ns
12 ns is common setting
No detector collimation required
Dual-head SPECT systems can be used
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 267 / 412
Emission Tomography PET System Components
PET Detector Block
Crystals plus PMTs
BGO = Bismuth Germanate
BGO has 3x stopping power than NaI(Tl)
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 268 / 412
Emission Tomography PET System Components
Typical PET Detector Arrangement2 mm × 2 mm elements
8 by 8 elements per blocks; 2 by 2 PMTs per block
48 blocks per major ring; 3 major rings
⇒ 24 detector rings; 384 detectors per ring
⇒ 8216 crystals total
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 269 / 412
Emission Tomography PET System Components
2-D or 3-D PET GeometrySepta or no septa between rings?
Septa: ⇒ multiple 2-D PET rings▶ Reconstruction like 2-D CT
No septa: ⇒ 3-D PET▶ Need 3-D reconstruction algorithms
We focus on 2-D PET
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 270 / 412
Emission Tomography PET Imaging Equation
2-D PET Geometry
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 271 / 412
Emission Tomography PET Imaging Equation
Lines of Response (LORs)
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 272 / 412
Emission Tomography PET Imaging Equation
Imaging EquationLine integrals of activity
On line L(ℓ, �)
�(ℓ, �) = K exp
{−∫ R
−R�(x(s), y(s);E ) ds
}×∫ R
−RA(x(s), y(s)) ds
Unknowns �(x , y) and A(x , y) separate
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 273 / 412
Emission Tomography PET Imaging Equation
Attenuation CorrectionCorrected sinogram
�c(ℓ, �) =�(ℓ, �)
K exp{−∫ R
−R �(x(s), y(s);E ) ds}
�(x , y) found from CT (transmission PET)
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 274 / 412
Emission Tomography PET Reconstruction
PET ReconstructionConvolution backprojection yields A(x , y)
Ac(x , y) =∫ �
0
∫ ∞−∞
�c(ℓ, �)c(x cos � + y sin � − ℓ) dℓd�
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 275 / 412
Emission Tomography Resolution and Artifacts
Resolution in Emission TomographyApproximation:
f (x , y) = f (x , y) ∗ h(r)
In SPECT, h(r) includes:▶ collimator and intrinsic resolutions▶ ramp filter window effect
In PET, h(r) includes:▶ the positron range function▶ detector width effects▶ ramp filter window effect
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 276 / 412
Emission Tomography Resolution and Artifacts
PET Events
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 277 / 412
Emission Tomography Resolution and Artifacts
Coincidence TimingThree classes of events
▶ true coincidence▶ scattered coincidence▶ random coincidence
Sensitivity in PET▶ measures capability of system to detect
“trues” and reject “randoms”
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Ultrasound Physics
10 Ultrasound PhysicsAn Ultrasound SystemWave EquationsWave PropagationField Patterns
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 279 / 412
Ultrasound Physics An Ultrasound System
Ultrasound Image
http://www.gehealthcare.com http://www.gehealthcare.com
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 280 / 412
Ultrasound Physics Wave Equations
UltrasoundUltrasound is sound with f > 20 kHz
Medical ultrasound imaging uses f > 1 MHz
Same physics = physics of longitudinal waves▶ Wave equations▶ Snell’s laws (reflection and refraction)▶ Attenuation and absorption▶ The Doppler effect▶ Vibrating plates and field patterns▶ Scattering
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 281 / 412
Ultrasound Physics Wave Equations
3-D Wave EquationAcoustic pressure: p(x , y , z , t)
3-D wave equation
∇2p(x , y , z , t) =1
c2ptt(x , y , z , t)
where∇2p = pxx + pyy + pzz
and c is the speed of sound
General solution is very complicated
We go after plane waves and spherical waves
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 282 / 412
Ultrasound Physics Wave Equations
Plane WavesPlane wave in z direction:
p(z , t) = p(x , y , z , t)
Plane wave equation:
pzz(z , t) =1
c2ptt(z , t)
General solution:
p(z , t) = �f (t − c−1z) + �b(t + c−1z)
where �f (t) and �b(t) are arbitrary
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 283 / 412
Ultrasound Physics Wave Equations
Harmonic Waves“Harmonic” plane wave
p(z , t) = cos[k(z − ct)]
Definitions:▶ wavenumber: k▶ frequency: f = kc/2�▶ period: T = 1/f▶ wavelength: � = c/f
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 284 / 412
Ultrasound Physics Wave Equations
Spherical Waves3-D spherical wave:
p(r , t) = p(x , y , z , t)
where r =√x2 + y 2 + z2.
Spherical wave equation:
1
r
∂2
∂r 2(rp) =
1
c2
∂2p
∂t2
General solution (outward expanding):
p(r , t) =1
r�o(t − c−1r)
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 285 / 412
Ultrasound Physics Wave Propagation
Characteristic ImpedanceCharacteristic impedance
Z = �c
where � is density
Why impedance?p = Zv
where v is particle velocity v ∕= c
▶ p is “like” voltage▶ v is “like” current
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 286 / 412
Ultrasound Physics Wave Propagation
Acoustic EnergyKinetic energy density:
wk =1
2�0v
2
Potential energy density:
wp =1
2�p2
where � is compressibility.
Acoustic energy density:
w = wk + wp
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 287 / 412
Ultrasound Physics Wave Propagation
Acoustic PowerAcoustic Intensity:
I = pv =p2
Z
(like electrical power p = vi)
Propagation of acoustic power (plane wave):
∂I
∂z+∂w
∂t= 0
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 288 / 412
Ultrasound Physics Wave Propagation
Reflection and Refraction
Snell’s Laws:
�r = �isin �isin �t
=c1
c2
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 289 / 412
Ultrasound Physics Wave Propagation
Reflected and Refracted WavesPressure reflectivity:
R =prpi
=Z2 cos �i − Z1 cos �tZ2 cos �i + Z1 cos �t
Pressure transmittivity:
T =ptpi
=2Z2 cos �i
Z2 cos �i + Z1 cos �t
At normal incidence:
R =Z2 − Z1
Z2 + Z1
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 290 / 412
Ultrasound Physics Wave Propagation
Attenuation and AbsorptionPhenomenological model:
p(z , t) = A0e−�az f (t − c−1z)
�a is amplitude attenuation factor [cm−1]
Absorption coefficient:
� = 20(log10 e)�a [dB/cm]
In range 1 MHz ≤ f ≤ 10 MHz
� ≈ af and a ≈ 1 dB/cm-MHz
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 291 / 412
Ultrasound Physics Wave Propagation
ScatteringParticle at (0, 0, d), reflection coefficient R
Generates spherical wave
ps(r , t) =Re−�arA0e
−�ad
rf (t − c−1d − c−1r)
r is distance from (0, 0, d)
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 292 / 412
Ultrasound Physics Field Patterns
Field PatternsGeometric approximation
Diffraction formulation (book)
Simple model:
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 293 / 412
Ultrasound Physics Field Patterns
Far Field = Fraunhofer PatternTransducer face indicator function:
s(x , y) =
{1 (x , y) in face0 otherwise
Far field pattern:
q(x , y , z) ≈ 1
ze jk(x2+y2)/2zS
( x
�z,y
�z
)S(u, v) is Fourier transform of s(x , y).
Pulse-echo sensitivity: q2(x , y , z).
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 294 / 412
Ultrasound Physics Field Patterns
FocusingFocal length field pattern:
q(x , y , d) ≈ 1
de jk(x2+y2)/2dS
( x
�d,y
�d
)
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 295 / 412
Ultrasound Imaging
11 Ultrasound ImagingUltrasound System ComponentsTransducersDisplay ModesEffects of AbsorptionPhased ArraysImaging EquationResolutionSpeckle
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Ultrasound Imaging Ultrasound System Components
Block Diagram
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Ultrasound Imaging Transducers
Transducerslead zirconate titantate (PZT)
▶ piezoelectric crystal▶ good transmit and receive efficiencies▶ different shapes:
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 298 / 412
Ultrasound Imaging Transducers
Piezoelectric Effect
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 299 / 412
Ultrasound Imaging Transducers
ResonanceShock excite yields resonant pulse
Resonant frequency:
fT =cT
2dT
Damps out after 3–5 cycles
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Ultrasound Imaging Transducers
Typical Transmit Pulse
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 301 / 412
Ultrasound Imaging Transducers
Ultrasound Probe
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Ultrasound Imaging Transducers
Mechanical Scanners
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 303 / 412
Ultrasound Imaging Transducers
Electronic Scanner
Linear arrays:▶ 64–256 elements, fire in groups▶ each element ≈ 2 mm by 10 mm
Phased arrays:▶ 30–128 elements; electronically steered▶ each element ≈ 0.2 mm by 8 mm
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 304 / 412
Ultrasound Imaging Display Modes
A-mode Display
The Range Equation
z =ct
2
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 305 / 412
Ultrasound Imaging Display Modes
M-mode Display
http://www.gehealthcare.com
fast time vs. slow timeJerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 306 / 412
Ultrasound Imaging Display Modes
B-mode Display
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 307 / 412
Ultrasound Imaging Effects of Absorption
Depth of PenetrationSignal is “lost” from absorption
Total travel before “lost” is
d =L
�
where L is system sensitivity in dB
depth of penetration is
dp =d
2=
L
2�≈ L
2af
Rule-of-thumb: dp ≈ 40/f (MHz) cm
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 308 / 412
Ultrasound Imaging Effects of Absorption
Pulse RepetitionSignal “dies”; then repeat
Pulse repetition interval:
TR ≥2dpc≈ L
afc
Pulse repetition frequency/rate:
fR =1
TR
@2 MHz, fR ≤ 3,850 Hz.
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 309 / 412
Ultrasound Imaging Effects of Absorption
Image Frame RateN scan lines to make image
Image frame rate:
fF ≤fRN
How to increase frame rate?▶ restrict field-of-view▶ increase frequency (why?)
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Ultrasound Imaging Phased Arrays
Phased Arrays: Transmit Steering
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 311 / 412
Ultrasound Imaging Phased Arrays
Phased Arrays: Transmit Focussing
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 312 / 412
Ultrasound Imaging Phased Arrays
Delays for Transmit FocussingFocal point at (xf , zf )
Ti is at (id , 0).
Then range from Ti to focal point is:
ri =√
(id − xf )2 + z2f
Assume T0 fires at t = 0. Then Ti fires at
ti =r0 − ri
c
=
√x2f + z2
f −√
(id − xf )2 + z2f
c
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Ultrasound Imaging Phased Arrays
Phased Arrays: Receive Beamforming
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 314 / 412
Ultrasound Imaging Phased Arrays
Phased Arrays: Receive DynamicFocussing
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Ultrasound Imaging Phased Arrays
Dynamic Focussing Time DelaysT0 fired at t = 0 (focussed or steered)
spherical wave originates at (x , z)
Distance from (x , z) to Ti is
ri =√
(id − x)2 + z2
Dynamic time delays are (requires derivation)
�i(t) = t −√
(id)2 + (ct)2 − 2ctid sin �
c+
Nd
c
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 316 / 412
Ultrasound Imaging Imaging Equation
Complex SignalComplex signal:
n(t) = ne(t)e j�e−j2�f0t
Complex envelope is n(t) = ne(t)e j�
The pulse isn(t) = Re{n(t)}
The envelope is
ne(t) = ∣n(t)∣
(This will form the A-mode signal.)
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 317 / 412
Ultrasound Imaging Imaging Equation
Complex Pressure in SpaceAcoustic dipole (complex) pressure
p(x , y , z ; t) =1
r0
z
r0n(t − c−1r0)
Superposition over transducer face
p(x , y , z ; t) =
∫ ∫s(x0, y0)
z
r 20
n(t − c−1r0)dx0dy0
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Ultrasound Imaging Imaging Equation
Echo from Point ScattererPoint scatterer with reflectivity R(x , y , z)
Pressure at (x ′0, y′0) on face
ps(x′0, y′0; t) = R(x , y , z)
1
r ′0p(x , y , z ; t − c−1r ′0)
Integrated dipole response (voltage)
r(x , y , z ; t) =
K
∫ ∫s(x ′0, y
′0)z
r ′0ps(x
′0, y′0; t)dx ′0dy
′0
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Ultrasound Imaging Imaging Equation
Total Response from Point Scatterer
r(x , y , z ; t) = KR(x , y , z)
⋅∫ ∫
dx ′0dy′0 s(x ′0, y
′0)
z
r ′20
⋅∫ ∫
dx0dy0 s(x0, y0)z
r 20
⋅ n(t − c−1r0 − c−1r ′0)
Now apply a series of approximations:▶ plane wave, paraxial▶ Fresnel, Fraunhofer
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 320 / 412
Ultrasound Imaging Imaging Equation
Plane Wave ApproximationExcitation pulse envelope arrives at all points at agiven range simultaneously.
Mathematically,
n(t − c−1r0 − c−1r ′0) ≈n(t − 2c−1z)e jk(r0−z)e jk(r ′0−z)
where wavenumber is
k = 2�f0c−1
and range equation gives
ct = 2z
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 321 / 412
Ultrasound Imaging Imaging Equation
Received Signal with Field PatternDefine field pattern as
q(x , y , z) =
∫ ∫s(x0, y0)
z
r 20
e jk(r0−z)dx0dy0
Then received signal (from single scatterer) is
r(x , y , z ; t) =
KR(x , y , z)n(t − 2c−1z)[q(x , y , z)]2
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 322 / 412
Ultrasound Imaging Imaging Equation
Basic Pulse-echo Imaging EquationSpatial distribution of scatterers
Assume superposition holds
Include attenuation
Total response is
r(t) = K
∫ ∫ ∫R(x , y , z)
n(t − 2c−1z)e−2�az [q(x , y , z)]2dxdydz
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 323 / 412
Ultrasound Imaging Imaging Equation
Paraxial ApproximationPattern is large near the transducer axis
Then r0 ≈ z
Field pattern becomes
q(x , y , z) ≈ 1
z
∫ ∫s(x0, y0)e jk(r0−z)dx0dy0
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 324 / 412
Ultrasound Imaging Imaging Equation
Fresnel and Fraunhofer ApproximationsBoth involve phase approximations
Fresnel field pattern
q(x , y , z) ≈ 1
zs(x , y) ∗ e jk(x2+y2)/2z
Fraunhofer field pattern
q(x , y , z) ≈ 1
ze jk(x2+y2)/2zS
( x
�z,y
�z
)for z ≥ D2/�.
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 325 / 412
Ultrasound Imaging Imaging Equation
General Pulse-echo EquationDefine
q(x , y , z) = zq(x , y , z)
Fresnel or Fraunhofer satisfies
r(t) = Ke−�act
(ct)2∫ ∫ ∫R(x , y , z)n(t − 2c−1z)q2(x , y , z)dxdydz
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 326 / 412
Ultrasound Imaging Imaging Equation
Time-gain CompensationAmplitude of r decays predictably
Compensate with time-varying gain
rc(t) = g(t)r(t) =∫ ∫ ∫R(x , y , z)n(t − 2c−1z)q2(x , y , z)dxdydz
g(t) =(ct)2e�act
K
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 327 / 412
Ultrasound Imaging Imaging Equation
Envelope Detection: A-modeComplex signal model n(t) throughout
Linear system model (superposition)
Therefore, gain-compensated A-mode signal is
ec(t) =
∣∣∣∣∫ ∫ ∫ R(x , y , z)
n(t − 2c−1z)q2(x , y , z)dxdydz∣∣
=
∣∣∣∣∫ ∫ ∫ R(x , y , z)
ne(t − 2c−1z)e j2kz q2(x , y , z)dxdydz∣∣
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 328 / 412
Ultrasound Imaging Imaging Equation
Transducer Motion and Range EquationMove transducer to (x0, y0); yields ec(t; x0, y0).
Use range equation as z0 = ct/2.
Then ec(⋅) estimates reflectivity
R(x0, y0, z0) = ec(2z0/c ; x0, y0)
=
∣∣∣∣∫ ∫ ∫ R(x , y , z)e j2kzne(2(z0 − z)/c)
⋅ q2(x − x0, y − y0, z)dxdydz∣∣
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 329 / 412
Ultrasound Imaging Resolution
Resolution CellWhere is the acoustic energy in space?
resolution cell(x , y , z ; x0, y0, z0) =
∣ne(2(z0 − z)/c)q2(x − x0, y − y0, z)∣
For geometric approximation
resolution cell(x , y , z ; x0, y0, z0) =
ne(2(z0 − z)/c)s(x − x0, y − y0)
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 330 / 412
Ultrasound Imaging Speckle
Origin of SpeckleUnder geometric assumption
R(x , y , z) =
K
∣∣∣∣R(x , y , z)e j2kz ∗ s(x , y)ne
(z
c/2
)∣∣∣∣Term e j2kz is “fast-changing” sinusoid in resolutioncell
Gives rise to essentially “random” constructive anddestructive interference
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 331 / 412
Physics of Magnetic Resonance
12 Physics of Magnetic ResonanceSpin SystemsMagnetizationNMR SignalExcitationRelaxationBloch EquationsSpin EchoesContrast
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 332 / 412
Physics of Magnetic Resonance Spin Systems
NucleiNMR is concerned with nuclei
... but not radioactivity
All nuclei have charge
Some nuclei have angular momentum Φ
Angular momentum + charge ≡ spin
Nuclei with spin are NMR-active
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 333 / 412
Physics of Magnetic Resonance Spin Systems
Visualization of Nuclear Spin
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 334 / 412
Physics of Magnetic Resonance Spin Systems
Nuclear Spin SystemsNuclear spin systems =
collections of identical nuclei▶ regardless of chemical environment▶ Examples: 1H, 13C, 19F, 31P
Whole-body MRI uses 1H▶ prevalent in the body (water, fat)▶ strong NMR signal▶ misnomer: “proton” imaging
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 335 / 412
Physics of Magnetic Resonance Magnetization
Microscopy Magnetic FieldMicroscopic magnetic moment vector:
� = Φ
is gyromagnetic ratio [radians/s-T]
– has more convenient units [Hz/T]
– =
2�
For 1H – = 42.58 MHz/T
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 336 / 412
Physics of Magnetic Resonance Magnetization
Nuclear MagnetismPut sample in external magnetic field
B0 = B0z
Spins align in one of two directions▶ 54∘ off z “up”▶ 180− 54∘ off z “down”
Slight preference for “up” direction
Sample becomes magnetized
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 337 / 412
Physics of Magnetic Resonance Magnetization
Macroscopic MagnetizationMagnetization vector:
M =
Ns∑n=1
�n
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 338 / 412
Physics of Magnetic Resonance Magnetization
Equilibrum MagnetizationEquilibrium value: M0
▶ same direction as B0
▶ depends on x = (x , y , z) only
Magnitude: M0
M0 =B0
2ℏ2
4k–TPD
▶ k– is Boltzmann’s constant▶ T is temperature▶ PD is proton density
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 339 / 412
Physics of Magnetic Resonance Magnetization
Evolution of MagnetizationM = M(x, t)
Relation to bulk angular momentum J
M = J
Focus on small sample → voxel
▶ M = M(t)▶ Equations of motion = Bloch equations
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 340 / 412
Physics of Magnetic Resonance Magnetization
Torque on “Current Loop”Current loop in magnetic field
▶ magnetic (dipole) moment M▶ magnetic field B▶ torque is � = M× B
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 341 / 412
Physics of Magnetic Resonance Magnetization
PrecessionTorque acts on rotating body in “funny” way
Torque is related to angular momentum
� =dJ
dt
Eliminate J to yield
dM(t)
dt= M(t)× B(t)
Equation describes precession
Valid for “short” times
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 342 / 412
Physics of Magnetic Resonance Magnetization
Larmor FrequencyLet B(t) = B0; M(0) angle � with z
Then
Mx(t) = M0 sin� cos (− B0t + �)
My(t) = M0 sin� sin (− B0t + �)
Mz(t) = M0 cos�
whereM0 = ∣M(0)∣ � arbitrary
Precession with Larmor frequency
!0 = B0 or �0 = –B0
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 343 / 412
Physics of Magnetic Resonance Magnetization
Isochromats!0 not constant for spin system due tomagnetic field inhomogeneities
Main magnetic field, shimming ⇒ ignore
magnetic susceptibility:▶ diamagnetic, paramagnetic, ferromagnetic▶ body/air interface strong change
chemical shift▶ chemical environment ⇒ shielding▶ fat is 3.35 ppm down from water
Isochromats: nuclei with same Larmor freq
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 344 / 412
Physics of Magnetic Resonance Magnetization
Magnetization ComponentsMagnetization
M(t) = (Mx(t),My(t),Mz(t))
Think of M(t) with two components
▶ Longitudinal magnetization
Mz(t)
▶ Transverse magnetization
Mxy(t) = Mx(t) + jMy(t)
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 345 / 412
Physics of Magnetic Resonance NMR Signal
Origin of NMR SignalPrinciple of Reciprocity
Br(r) is field produced at r by unit directcurrent in coil around sample.
⇒ Now reverse scenario ⇐Voltage produced in coil by changing magnetic fieldis (by Faraday’s law of induction)
V (t) = − ∂
∂t
∫object
M(r, t) ⋅ Br(r) dr
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 346 / 412
Physics of Magnetic Resonance NMR Signal
NMR SignalLongitudinal magnetization changes too slow
Transverse magnetization dominates
Mxy(t) = M0 sin�e−j(!0t−�)
Final expression
V (t) = −!0VsM0 sin�B r sin(−!0t + �− �r)
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 347 / 412
Physics of Magnetic Resonance NMR Signal
Rotating FrameCoordinate transformation
x ′ = x cos(!0t)− y sin(!0t)
y ′ = x sin(!0t) + y sin(!0t)
z ′ = z
Transverse magnetization in rotating frame
Mx ′y ′(t) = M0 sin�e j�
Magnitude M0 sin�
Phase angle �
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 348 / 412
Physics of Magnetic Resonance Excitation
RF ExcitationCircularly polarized RF excitation pulse
B1(t) = Be1 (t)e−j(!0t−')
Yields forced precession
M(t) motion is spiral
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 349 / 412
Physics of Magnetic Resonance Excitation
Tip Anglez-magnetization magnitude after excitation
Mz = M0 cos�
Tip angle is
� =
∫ �p
0
Be1 (t)dt
where �p is pulse duration
For rectangular pulse:
� = B1�p
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 350 / 412
Physics of Magnetic Resonance Relaxation
RelaxationMagnetization cannot precess forever
Two independent relaxation processes
Transverse relaxation▶ ≡ spin-spin relaxation
Longitudinal relaxation▶ ≡ spin-lattice relaxation
Detailed properties differ in tissues▶ Gives rise to tissue contrast
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 351 / 412
Physics of Magnetic Resonance Relaxation
Transverse Relaxation
Transverse relaxation decays
Mxy(t) = M0 sin�e−j(!0t−�)e−t/T2
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 352 / 412
Physics of Magnetic Resonance Relaxation
Free Induction DecayWhat RF signal is produced?
Called a free induction decay (FID)
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 353 / 412
Physics of Magnetic Resonance Relaxation
T ∗2 DecayIn fact RF signal decays faster
T ∗2 < T2
Underlying T2 relaxation is preservedJerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 354 / 412
Physics of Magnetic Resonance Relaxation
Longitudinal RelaxationMz(t) behaves as rising exponential
Mz(t) = M0(1− e−t/T1) + Mz(0+)e−t/T1
Mz(0+) is value after RF excitation pulseM0 is final (equilibrium) value
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 355 / 412
Physics of Magnetic Resonance Bloch Equations
Bloch EquationsEquation(s) of “motion” for M(t)
dM(t)
dt= M(t)× B(t)− R{M(t)−M0}
Includes RF excitation
B(t) = B0 + B1(t) ,
Includes relaxation
R =
⎛⎝ 1/T2 0 00 1/T2 00 0 1/T1
⎞⎠Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 356 / 412
Physics of Magnetic Resonance Spin Echoes
Concept of Spin EchoesPure transverse relaxation T2 is random
So why faster decay T ∗2 ?
▶ Fixed, local perturbations in magnetic field▶ Local dephasing from faster & slower spins
Echoes produced by “re-phasing”
▶ Make slower spins “jump” to front▶ Make faster spins “jump” to rear
TE is echo time
Multiple echoes are possible until about 3T2
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 357 / 412
Physics of Magnetic Resonance Spin Echoes
Formation of a Spin Echo
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 358 / 412
Physics of Magnetic Resonance Spin Echoes
Pulse Sequence for Spin Echo
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 359 / 412
Physics of Magnetic Resonance Contrast
Source of MR ContrastMR Contrast: why tissues “look” different in MRI
Intrinsic MR parameters:▶ T1, T2, and PD
Pulse sequence parameters:▶ tip angle �▶ echo time TE
▶ pulse repetition interval TR
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 360 / 412
Physics of Magnetic Resonance Contrast
Contrast Manipulation
PD-weighted T2-weighted T1-weighted
Weighted means “primarily influenced by”
Weighted does not mean “a picture of”
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 361 / 412
Physics of Magnetic Resonance Contrast
PD-weighted ContrastShould be proportional to # 1H nuclei in voxel
Procedure:▶ Start with sample in equilibrium▶ Apply excitation pulse▶ Image quickly
Practical parameters:▶ TR = 6000 ms (long)▶ TE = 17 ms (relatively quick)▶ � = �/2 (max signal)
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 362 / 412
Physics of Magnetic Resonance Contrast
T2-weighted ContrastEcho must be used because of T ∗2Procedure:
▶ Start with sample in equilibrium▶ Apply excitation pulse▶ Image at approx T2
Practical parameters:▶ TR = 6000 ms (long)▶ TE = 102 ms (moderate)▶ � = �/2 (max signal)
Trick: get PD and T2 in two echoes
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 363 / 412
Physics of Magnetic Resonance Contrast
Principle of T1-weighted Contrast
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 364 / 412
Physics of Magnetic Resonance Contrast
T1-weighted ContrastUses TR to capture T1 differences
Procedure:▶ Reach a steady-state, not equilibrium▶ Apply excitation so that TR ≈ T1
▶ Image quickly
Possible parameters:▶ TR = 600 ms (moderate)▶ TE = 17 ms (fast)▶ � = �/2 (max signal)
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 365 / 412
Magnetic Resonance Imaging
13 Magnetic Resonance ImagingMR Scanner ComponentsFrequency EncodingSlice SelectionSignal ModelsScanning Fourier SpaceGradient EchoesPhase EncodingSpin EchoesRealistic Pulse SequencesImage ReconstructionSampling, Resolution, and Noise
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 366 / 412
Magnetic Resonance Imaging MR Scanner Components
Five System Components1 Main magnet2 Gradient coils3 RF resonators or coils4 Pulse sequence electronics5 Computer and viewing console
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 367 / 412
Magnetic Resonance Imaging MR Scanner Components
MR Scanner Components
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 368 / 412
Magnetic Resonance Imaging MR Scanner Components
MR Scanner Photograph
http://www.gehealthcare.com http://www.gehealthcare.com
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 369 / 412
Magnetic Resonance Imaging MR Scanner Components
Superconducting Magnet1 meter niobium-titanium wiresuperconducting coils4∘ K liquid helium cryostat
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 370 / 412
Magnetic Resonance Imaging MR Scanner Components
Magnet SpecificationsField strengths from 0.5T to 12.0T
Most common field strength: 1.5T
Shimming to maintain homogeneous field▶ passive shimming▶ active shimming▶ better than ±5 ppm required
Minimize fringe field (outside the bore)
▶ nuisance and dangerous▶ passive: iron shield, or▶ active: second superconducting wires
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 371 / 412
Magnetic Resonance Imaging MR Scanner Components
Purpose of Gradient CoilsFit just inside the bore
Role: change B0 as a function of position
Three coils:▶ x , y , and z directions▶ Gx , Gy , and Gz strengths
Modify main field as follows
B = (B0 + Gxx + Gyy + Gzz)z
This is the key to MR imaging
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 372 / 412
Magnetic Resonance Imaging MR Scanner Components
Gradient Coilsx and y are saddle coilsz is opposing coils
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 373 / 412
Magnetic Resonance Imaging MR Scanner Components
Specifications of Gradient CoilsMaximum gradient 1–6 Gauss/cm
Switching times 0.1–1.0 ms
slew rates 5–250 mT/m/msec
Additional shielding outside to reduceeddy currents
FDA limit 40 T/s
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 374 / 412
Magnetic Resonance Imaging MR Scanner Components
RF CoilsTwo purposes:
▶ Exciting spin systems▶ Listening for FIDs and echoes
Two basic types:▶ volume coils▶ surface coils
Volume coils have uniform response
Surface coils are more sensitive buthave spatially-dependent response
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 375 / 412
Magnetic Resonance Imaging MR Scanner Components
RF Coil Designs
(a) saddle coil: head
(b) birdcage coil: body, head
(c) surface loop: peripherals
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 376 / 412
Magnetic Resonance Imaging MR Scanner Components
Scanning Console and ComputerControl scanner
Acquire images
Coordinate with EKG and breathing
Reconstruct images (10–50 images/s)
View, store, and manipulate images
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 377 / 412
Magnetic Resonance Imaging MR Scanner Components
Laboratory Coordinates
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 378 / 412
Magnetic Resonance Imaging Frequency Encoding
(Larmor) Frequency EncodingGradient G = (Gx ,Gy ,Gz) produces B-field:
B = (B0 + G ⋅ r)z
where r = (x , y , z)
Spatially varying Larmor frequency
�(r) = –(B0 + G ⋅ r)
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 379 / 412
Magnetic Resonance Imaging Slice Selection
Principle of Slice SelectionLet G = (0, 0,Gz)
Then�(r) = �(z) = –(B0 + Gzz)
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 380 / 412
Magnetic Resonance Imaging Slice Selection
Slice Selection ExcitationExcite frequencies � ∈ [�1, �2]
Causes “slab” excitation of spin system
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 381 / 412
Magnetic Resonance Imaging Slice Selection
Slice Selection ParametersRF parameters:
� =�1 + �2
2center frequency
Δ� = ∣�2 − �1∣ frequency range
Slice parameters:
z =� − �0
–Gzslice position
Δz =Δ�
–Gzslice thickness
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 382 / 412
Magnetic Resonance Imaging Slice Selection
Ideal Slice Selection RF ExcitationExcite frequencies in range [�1, �2] HzExcitation signal has Fourier transform
S(�) = A rect
(� − �
Δ�
)Signal is s(t) = A� sinc(�t)e j2��t
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 383 / 412
Magnetic Resonance Imaging Slice Selection
Practical Slice Selection RF ExcitationTruncated sinc
s(t) =[A� sinc(�t)e j2��t
]rect(t/�p)
Corresponding tip angle profile:
�(z) = A�prect
(z − z
Δz
)∗ sinc (�p Gz(z − z))
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 384 / 412
Magnetic Resonance Imaging Slice Selection
Slice Dephasing and RefocussingDifference Larmor frequencies across slice:
▶ “slow” on low side▶ “fast” on high side
Phase difference is
�(z) = Gz(z − z)�p/2
Refocus with negative gradient pulse▶ strength −Gz
▶ duration �/2
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 385 / 412
Magnetic Resonance Imaging Slice Selection
A Simple Pulse Sequence
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 386 / 412
Magnetic Resonance Imaging Signal Models
Basic Signal ModelSlice selection FID is
s(t) = e−j2��0t
∫ ∞−∞
∫ ∞−∞
f (x , y) dx dy
Where effective spin density is
f (x , y) = AM(x , y ; 0+)e−t/T2(x ,y)
Baseband signal is
s0(t) =
∫ ∞−∞
∫ ∞−∞
f (x , y) dx dy
Note: T ∗2 always decays the FID signal
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 387 / 412
Magnetic Resonance Imaging Signal Models
Frequency Encoding
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 388 / 412
Magnetic Resonance Imaging Signal Models
Frequency Encoding SignalLarmor frequency is function of x
�(x) = –(B0 + Gxx)
Baseband signal becomes
s0(t) =
∫ ∞−∞
∫ ∞−∞
f (x , y)e−j2� –Gxxt dx dy
Recognize Fourier transform frequencies:
u = –Gxt
v = 0
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 389 / 412
Magnetic Resonance Imaging Scanning Fourier Space
Relation to Fourier transform
F (u, 0) = s0
(u
–Gx
)
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 390 / 412
Magnetic Resonance Imaging Scanning Fourier Space
Polar Scanning
u = –Gxt v = –Gy t
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 391 / 412
Magnetic Resonance Imaging Scanning Fourier Space
Scanning Fourier SpaceIgnore readout/ADC
Applied gradients “drive” us around in Fourier space
This is concept of Fourier trajectory
Fourier trajectories underly all MRI▶ spin echoes▶ gradient echoes▶ frequency encoding▶ polar scanning▶ phase encoding
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 392 / 412
Magnetic Resonance Imaging Gradient Echoes
Gradient Echoes
Note: T ∗2 decay overall
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 393 / 412
Magnetic Resonance Imaging Phase Encoding
Phase Encoding
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 394 / 412
Magnetic Resonance Imaging Phase Encoding
Phase Encoding SignalAccumulated phase after phase encode
�y(y) = − GyTpy
Baseband signal during readout
s0(t) =
∞∫−∞
∞∫−∞
f (x , y)e−j2� –Gxxte−j2� –GyTpy dx dy
Recognize Fourier transform frequencies:
u = –Gxt
v = –GyTp
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 395 / 412
Magnetic Resonance Imaging Phase Encoding
Gradient Echo Pulse Sequence
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 396 / 412
Magnetic Resonance Imaging Spin Echoes
Concept of a Spin Echo
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 397 / 412
Magnetic Resonance Imaging Spin Echoes
Basic “Spin Echo” Pulse Sequence
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 398 / 412
Magnetic Resonance Imaging Realistic Pulse Sequences
Realistic Gradient Echo Pulse Sequence
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 399 / 412
Magnetic Resonance Imaging Realistic Pulse Sequences
Realistic Spin Echo Pulse Sequence
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 400 / 412
Magnetic Resonance Imaging Realistic Pulse Sequences
Realistic Spin Echo Polar Pulse Sequence
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 401 / 412
Magnetic Resonance Imaging Image Reconstruction
Acquired Rectilinear DataAcquire data for all phase encode areas
Ay = GyTp
Baseband signal
s0(t,Ay) =
∫ ∫f (x , y)e−j2� –Gxxte−j2� –Ayydxdy
Identify Fourier frequencies
u = –Gxt
v = –Ay
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 402 / 412
Magnetic Resonance Imaging Image Reconstruction
Image Reconstruction: Rectilinear DataFourier transform is built over repetitions
F (u, v) = s0
(u
–Gx,v
–
)0 ≤ u ≤ –GxTs
Inverse Fourier transform
f (x , y) =
∫ ∫s0
(u
–Gx,v
–
)e+j2�(ux+vy)dxdy
This is a fundamental equation in MRI
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 403 / 412
Magnetic Resonance Imaging Image Reconstruction
Acquired Polar DataGx and Gy identify readout parameters
% = –t√
G 2x + G 2
y
� = tan−1 Gy
Gx
Projection slice theorem connects 2D Fourier spaceto Fourier transform of 1D projection
G (%, �) = F (% cos �, % sin �)
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 404 / 412
Magnetic Resonance Imaging Image Reconstruction
Image Reconstruction: Polar DataBaseband signal is s0(t, �)
Relation to 1-D projection
G (%, �) = s0
⎛⎜⎝ %
–√G 2x + G 2
y
, �
⎞⎟⎠Filtered backprojection is the answer
f (x , y) =∫ �
0
[∫∣%∣G (%, �)e j2�%ℓ d%
]ℓ=x cos �+y sin �
d�
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 405 / 412
Magnetic Resonance Imaging Sampling, Resolution, and Noise
SamplingDuration of readout
Ts = NaT
Receiver bandwidth (sampling rate)
fs =1
T
Antialiasing filter chops outside [−fs/2, fs/2]
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 406 / 412
Magnetic Resonance Imaging Sampling, Resolution, and Noise
Readout Field of ViewAntialiasing filter chops Larmor frequencies,leading to
FOVx =fs –Gx
=1
–GxT
Step in Fourier space is
Δu = –GxT
Relation to field of view
FOVx =1
Δu
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 407 / 412
Magnetic Resonance Imaging Sampling, Resolution, and Noise
Phase Encode Field of ViewStep size in phase encode direction:
Δv = – ΔAy
Field of view
FOVy =1
– ΔAy
=1
Δv
Lack of antialising filter could cause wrap-around
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 408 / 412
Magnetic Resonance Imaging Sampling, Resolution, and Noise
ResolutionFourier space coverage
U = Nx –GxT
V = Ny –ΔAy
Implied lowpass filter is
H(u, v) = rect( uU
)rect
( vV
)Spatial PSF is
h(x , y) = UV sinc(Ux)sinc(Vy)
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 409 / 412
Magnetic Resonance Imaging Sampling, Resolution, and Noise
Full Width Half Max’sFWHMs are
FWHMx =1
U=
1
Nx –GxT=
1
NxΔu
FWHMy =1
V=
1
Ny –ΔAy=
1
NyΔv
The Fourier resolutions of MRI
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 410 / 412
Magnetic Resonance Imaging Sampling, Resolution, and Noise
NoiseJohnson (thermal) noise dominates
�2 =2k–T RTA
k– = Boltzmann’s constant
T = temperature ⇒ colder is better
R = effective resistance ⇒ use small coils
TA = total acquisition time ⇒ scan longer
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 411 / 412
Magnetic Resonance Imaging Sampling, Resolution, and Noise
Signal-to-Noise RatioRecall magnitude of signal is
∣V ∣ = 2��0VsM0 sin�B r
Signal-to-noise Ratio is
SNR =∣V ∣√�2
= h2
√4�k–
2��0PD√�
r 20
√LT 3
Vs sin�√TA
Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 412 / 412